CN110501177A - Damage identification method for cantilever beams based on the curvature of free end inclination influence line - Google Patents

Abstract

The invention discloses a damage identification method for a cantilever beam based on the curvature of the influence line of inclination angle at the free end. The steps are as follows: applying a moving load to the damaged cantilever beam to obtain the actually measured inclination angle influence line at the measuring point at the free end; The curvature of the influence line is calculated; the stiffness of each position of the beam structure is obtained by dividing the load value by the curvature of the influence line of the inclination angle, and the damage position is identified through the sudden change of the stiffness curve of the damage state; the stiffness of the damage position is eliminated, and the remaining stiffness curve is fitted to obtain the undamaged state The stiffness curve of the damage and undamaged state is used to calculate the damage degree, and the structural stiffness of the damage position is obtained. The invention can accurately locate and quantify the damage of the cantilever beam, and is applied to the damage assessment of the cantilever beam. The application value of the curvature index of the influence line of inclination angle in the damage identification of the cantilever beam is verified through the calculation examples of the cantilever beam with constant section and variable section.

Description

Damage identification method for cantilever beams based on the curvature of free end inclination influence line

technical field

The invention relates to the technical field of non-destructive testing of beam structures, in particular to a damage identification method of a cantilever beam based on the curvature of a free end inclination influence line.

Background technique

In recent years, there are more and more old bridges in our country, and the problems are becoming more and more obvious. Many existing bridges can no longer meet the functional requirements, and safety accidents such as bridge breakage and collapse occur from time to time. Scholars in the field of civil engineering have gradually realized the importance of health monitoring and safety assessment of bridge structures, and have studied various damage identification technologies. . Structural damage identification is an important part of the bridge structural health monitoring system. At present, there are two main types of damage identification methods. One is the damage identification method based on dynamic parameters, which is mainly judged by the change of structural mode (vibration frequency and mode shape). Structural damage, such methods have high requirements on the number of measuring points, sensor measurement accuracy, and modal parameter identification methods. Another type of method is the damage identification method based on static parameters. The structural damage identification method based on static parameters can effectively avoid the uncertain influence of quality, especially damping. Quite accurate measurement values of the structure can be obtained at a relatively low cost. Therefore, the structural damage identification technology based on static parameters has been extensively studied.

The indicators of structural damage identification technology based on static parameters are mostly based on deflection, static strain, and support reaction force influence line indicators. With the advancement of inclination sensor technology, the change of structural inclination curve before and after damage is expected to be applied to At present, there are few literature reports on the damage identification of dip angle.

Contents of the invention

In order to solve the above-mentioned technical problems, the present invention provides a cantilever beam damage identification method based on the free end inclination influence line curvature with simple algorithm and low cost.

The technical solution of the present invention to solve the above-mentioned problems is: a method for identifying damage of a cantilever beam based on the curvature of the influence line of the inclination angle of the free end, the steps are as follows:

(1) Apply a moving load to each measuring point of the damaged cantilever beam to obtain the measured inclination influence line of the free end measuring point;

(2) Calculate the curvature of the measured inclination influence line after the beam structure is damaged;

(3) The stiffness of each position of the beam structure is obtained by dividing the load by the curvature of the influence line of the inclination angle, and the damage position is identified by the sudden change of the stiffness curve in the damage state;

(4) Eliminate the stiffness at the damaged position, and fit the remaining stiffness curve to obtain the stiffness curve in the undamaged state;

(5) Calculate the damage degree from the stiffness curves of the damaged and undamaged states, and obtain the structural stiffness of the damaged position.

In the above-mentioned cantilever beam damage identification method based on the curvature of the free end inclination angle influence line, in step (2), the inclination angle influence line curvature θ″ is calculated by the center difference:

Among them, the subscript i is the number of the measuring point, θ _i ″ is the curvature of the slope influence line of the measuring point i, ε is the average value of the distance from the measuring point i-1 to the measuring point i and the distance from the measuring point i to the measuring point i+1 , θi is the inclination angle of the test position when the load acts on the _i measuring point.

In the above-mentioned cantilever beam damage identification method based on the curvature of the free end inclination influence line, in step (3), the calculation method of the structural damage state stiffness curve B _d is:

Among them, B _di is the stiffness of the damaged state of the i measuring point, P is the moving load value, θ _i ″ is the curvature of the inclination influence line of the load acting on the i measuring point, n is the number of measuring points, and the 1st measuring point is arranged on the cantilever beam At the fixed support end, n measuring points are arranged at the free end of the cantilever beam. The number of measuring points is continuous, increasing from 1 to n, and i is greater than or equal to 2 and less than or equal to n-1.

In the above-mentioned cantilever beam damage identification method based on the curvature of the influence line of the inclination angle of the free end, in step (3), if the sudden change of the stiffness curve is not obvious, the slope of the stiffness curve is further calculated to assist in judging the damage location. The calculation method of the slope of the stiffness curve is as follows:

Among them, B′ _dj is the slope of the stiffness curve in the damage state of measuring point j, and ε ^* is the distance from measuring point j-1 to measuring point j.

In the above-mentioned cantilever beam damage identification method based on the curvature of free end inclination influence line, in step (4), linear fitting is used for the stiffness curve of the constant-section beam in the undamaged state, and local parabolic fitting is used for the variable-section beam. The fitting stiffness curve B _u is:

B _u = [0 B _u2 ... B _ui ... B _u(n-1) 0];

Among them, B _ui is the stiffness of the undamaged state fitted to the i-th measuring point.

In the above-mentioned cantilever beam damage identification method based on the curvature of the influence line of the inclination angle of the free end, in step (5), the calculation method of the quantitative index D _e of the structural damage degree is:

D _e = [0 D _e2 ... D _ei ... D _e(n-1) 0];

Among them, _{De ei} is the structural damage degree identified by the i-th measuring point, and the calculation method is:

In the above-mentioned cantilever beam damage identification method based on the curvature of the free end inclination angle influence line, in step (1), there are no less than six measurement points of the structure inclination angle influence line.

The beneficial effect of the present invention is: firstly, the present invention applies moving load to the cantilever beam after damage, obtains the inclination influence line of free end measuring point after beam damage, then calculates the curvature of the actually measured inclination influence line after beam structure damage, and then calculates the curvature by the load Divide by the curvature of the inclination influence line to obtain the stiffness of the beam structure in the damaged state, judge the damage position according to the sudden change of the stiffness curve of the damage state, then remove the stiffness of the damage position, and fit the remaining stiffness curve to obtain the stiffness curve of the beam structure before damage, and finally by the damage The stiffness curve of the front beam structure and the stiffness curve of the damaged state are used to calculate the damage degree. The invention verifies the application value of the curvature index of the inclination angle influence line in the damage identification of the cantilever beam through the calculation examples of the cantilever beam with constant section and variable section, and provides an effective new method for the damage location, quantification and stiffness identification of the cantilever beam.

Description of drawings

Fig. 1 is a block flow diagram of the method of the present invention.

Fig. 2 is a structural model diagram of the cantilever beam of the present invention.

Fig. 3 is a bending moment diagram of unit bending moment at the free end of the cantilever beam of the present invention.

Fig. 4 is a bending moment diagram of the unit force acting on the measuring point i-1 of the cantilever beam of the present invention.

Fig. 5 is a bending moment diagram of the unit force acting on the measuring point i of the cantilever beam of the present invention.

Fig. 6 is a bending moment diagram of unit force acting on the measuring point i+1 of the cantilever beam of the present invention.

Fig. 7 is a finite element model diagram of a cantilever beam with constant cross-section according to an embodiment of the present invention.

Fig. 8 is a schematic diagram of the inclination influence line of the damage state measuring point 21 in the first embodiment of the present invention.

Fig. 9 is a schematic diagram of the curvature of the inclination influence line of the damage state measuring point 21 in the first embodiment of the present invention.

Fig. 10 is a schematic diagram of the stiffness curve identified by the measuring point 21 in the first embodiment of the present invention.

Fig. 11 is a schematic diagram of the structural damage degree De _identified by the measuring point 21 in the first embodiment of the present invention.

Fig. 12 is a finite element model diagram of a cantilever beam with variable cross-section according to the second embodiment of the present invention.

Fig. 13 is a schematic diagram of the inclination influence line of the damage state measuring point 21 in the second embodiment of the present invention.

Fig. 14 is a schematic diagram of the curvature of the inclination influence line of the damage state measuring point 21 in the second embodiment of the present invention.

Fig. 15 is a schematic diagram of the stiffness curve identified by the measuring point 21 in the second embodiment of the present invention.

Fig. 16 is a schematic diagram of the slope of the stiffness curve identified by the measuring point 21 in the second embodiment of the present invention.

Fig. 17 is a schematic diagram of stiffness curve fitting of measuring points 3-5 in Embodiment 2 of the present invention.

Fig. 18 is a schematic diagram of stiffness curve fitting of measuring points 8, 9, 12 and 13 in the second embodiment of the present invention.

Fig. 19 is a schematic diagram of stiffness curve fitting of measuring points 17-19 in the second embodiment of the present invention.

Fig. 20 is a schematic diagram of the structural damage degree De _identified by the measuring point 21 in the second embodiment of the present invention.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

As shown in Figure 1, a damage identification method for a cantilever beam based on the curvature of the free end inclination influence line, the specific steps are as follows:

Step 1: Apply a moving load to each measuring point of the damaged cantilever beam to obtain the measured inclination influence line of the free end measuring point.

In step 1, the structural model of the cantilever beam is shown in Figure 2. The span of the cantilever beam is L, and A is the left end point of the cantilever beam. The distance between the damage location and the left end A is a, and the distance between each measuring point is ε. The stiffness of the damaged structure is EI, and the stiffness of the damaged element is EI _d . The bending moment when the unit bending moment M=1 acts on the free end position is (as shown in Figure 3):

Bending moments M ₁ , M ₂ , and M ₃ when load P acts on measuring points i-1, i, and i+1 respectively (as shown in Figures 4 to 6) are:

M ₁ =P(ax) (2)

M ₂ =P(a+ε-x) (3)

M ₃ ＝P(a+2ε-x) (4)

In the formula, x represents the distance from the left end point A of the cantilever beam.

Therefore, when the structure is damaged, the inclination angles of each measuring point can be obtained as follows:

In the formula, θ _id represents the inclination angle of measuring point i when the structure is damaged, and the subscript “d” represents the state of structural damage.

Step 2: Calculate the curvature of the measured inclination influence line after the beam structure is damaged.

In step 2, the curvature θ″ _id of the inclination influence line of the measurement point i on the right side of the damage position is calculated by using the central difference method as:

Step 3: Divide the load by the curvature of the inclination influence line to obtain the stiffness of each position of the structure, and identify the damage position through the sudden change of the stiffness curve in the damage state.

In step 3, the calculation method of the structural damage state stiffness curve B _d is:

Among them, B _di is the stiffness of the damage state of the i measuring point, P is the moving load value, θ″ _id is the curvature of the inclination influence line of the load acting on the i measuring point, n is the number of measuring points, and the 1st measuring point is arranged on the cantilever beam At the fixed support end, n measuring points are arranged at the free end of the cantilever beam. The number of measuring points is continuous, increasing from 1 to n, and i is greater than or equal to 2 and less than or equal to n-1.

When identifying the damage position through the sudden change of the stiffness curve in the damage state, if the sudden change of the stiffness curve is not obvious, the slope of the stiffness curve can be further calculated to assist in judging the damage position. The calculation method for the slope of the stiffness curve is as follows:

Among them, B′ _dj is the slope of the stiffness curve in the damage state of measuring point j, and ε ^* is the distance from measuring point j-1 to measuring point j.

Step 4: Eliminate the stiffness at the damaged position, and fit the remaining stiffness curve to obtain the stiffness curve in the undamaged state.

In step 4, the linear fitting is used for the stiffness curve of the undamaged state of the constant cross-section beam, and the local parabolic fitting is used for the variable cross-section beam. The fitted stiffness curve B _u of the undamaged state is:

B _u ＝[0 B _u2 ... B _ui ... B _u(n-1) 0] (11)

Among them, B _ui is the stiffness of the undamaged state fitted to the i-th measuring point.

Step 5: Calculate the damage degree from the stiffness curves of the damaged and undamaged states, and obtain the structural stiffness of the damaged position.

In step 5, when measuring point i-1, when the unit between i is not damaged, EI=B _ui .

When the unit between measuring point i-1, i is damaged:

Substituting formula (12) into formula (8) can be obtained:

Therefore, the damage degree of the unit can be obtained as:

In step 1, there are no less than 6 measurement points on the structure dip angle influence line.

Embodiment 1: Referring to Fig. 7, the span of a cantilever beam with equal section is 100cm, and a unit is divided by 5cm, a total of 20 units and 21 measuring points (the numbers in the circles in the upper row in the figure are unit numbers, and the numbers in the lower row are measuring points Numbering). The cross-sectional size of the plate is b×h=4.5cm×1.5cm, the elastic modulus of the material is 2.7×10 ³ MPa, the Poisson’s ratio is 0.37, and the density is 1200kg/m ³ .

The damage in the actual engineering structure, such as the generation of cracks, material corrosion or the reduction of elastic modulus, generally only causes a large change in the structural stiffness, but has little impact on the quality of the structure. Therefore, in the finite element calculation, it is assumed that the damage of the structural element only causes the decrease of the stiffness of the element, and does not cause the change of the mass of the element. Damage to elements is simulated by a reduction in the elastic modulus. The beam structure model is established by ANSYS software beam3 beam element. Taking the multi-unit damage condition as an example, considering that the fixed support end unit 1, the mid-span unit 10, and the free end unit 20 have different degrees of damage, the damage conditions are shown in Table 1.

Table 1 Multiple damage conditions of the cantilever beam

The specific implementation steps are as follows:

Step 1: Apply a moving load of 1kN to obtain the measured inclination influence line of the free end measuring point 21 after the cantilever beam is damaged, as shown in Figure 8 respectively.

Step 2: Calculate the curvature of the inclination influence line after the beam structure is damaged, as shown in Figure 9, the results show that there are obvious peaks at units 1, 10, and 20, and it can be preliminarily judged that units 1, 10, and 20 are damaged.

Step 3: Divide the load value by the curvature value of the inclination influence line to obtain the structural stiffness curve in the damaged state as shown in Figure 10. It can be seen that the stiffness of elements 1, 10, and 20 has dropped significantly, which is the damaged element.

Step 4: Eliminate the stiffness values of the measuring points around unit 1, 10, and 20, and perform linear fitting on the remaining stiffness curve to obtain a constant stiffness of the beam structure without damage, which is about 34.172N·m ² .

Step 5: Calculate the damage degree from the stiffness curves of the damaged and undamaged states, as shown in Figure 11, the identified damage degree is basically the same as the theoretical value, and then the actual stiffness EI _d of the damaged element can be calculated. For example, when the damage degree is 0.3, the damage The stiffness of the unit is EI _d =EI(1-0.3)=34.172*0.7=23.92N·m ² , which is smaller than that shown in FIG. 10 .

Embodiment 2: Referring to Figure 12, the span of the variable-section cantilever beam is 100cm, the division of units and nodes is the same as in Embodiment 1, the free end of the plate section size is b×h=4.5cm×1.5cm, and the fixed support end is b×h =22.5cm×3.0cm, the elastic modulus of the material is 2.7×10 ³ MPa, the Poisson’s ratio is 0.37, and the density is 1200kg/m ³ .

The damage condition is the same as that in Embodiment 1.

The specific implementation steps are as follows:

Step 1: Apply a moving load of 1kN to obtain the influence line of the measured inclination angle of the free end measuring point 21 after the cantilever beam is damaged, as shown in Figure 13 respectively.

Step 2: Calculate the curvature of the inclination influence line after the damage of the beam structure, as shown in Figure 14, the result shows that there is an obvious peak at unit 10, and it can be preliminarily judged that unit 10 is damaged, and the mutation at other positions is not obvious.

Step 3: Divide the load value by the curvature value of the inclination influence line to obtain the structural stiffness curve in the damaged state as shown in Figure 15. It can be seen that the stiffness of elements 1 and 10 has dropped significantly, which is the damaged element. In order to further judge whether there are other damaged elements, the Calculate the slope of the stiffness curve, as shown in Figure 16, it can be seen that the free end unit 20 may also be damaged, so it is judged that the units 1, 10, and 20 are damaged.

Step 4: Use piecewise parabola fitting for the stiffness curve, and fit the stiffness values of measuring points 3 to 5 as shown in Figure 17, take the serial number = 0, and obtain the stiffness fitting value of measuring point 2 in the undamaged state as 1216.5N· m ² , fit the stiffness values of measuring points 8, 9, 12, and 13, and obtain the undamaged stiffness values of measuring points 10 and 11, as shown in Figure 18, fit the stiffness values of measuring points 17 to 19, and obtain The undamaged stiffness value of measuring point 20, as shown in Figure 19, finally obtains the stiffness curve of the undamaged beam structure.

Step 5: Calculate the damage degree from the stiffness curves of the damaged and undamaged states, as shown in Figure 20, the recognized damage degree is close to the theoretical value, and the recognized damage degree of the free end is slightly larger than the theoretical value, which is 0.330, which is 10% larger. The error is acceptable for engineering applications.

The above descriptions are only two embodiments of the present invention, and all equivalent changes and modifications made according to the scope of the patent application of the present invention belong to the scope of the present invention.

Claims (7)

1. A cantilever beam damage identification method based on free end inclination influence line curvature, is characterized in that, comprises the steps: (1) Apply a moving load to each measuring point of the damaged cantilever beam to obtain the measured inclination influence line of the free end measuring point; (2) Calculate the curvature of the measured inclination influence line after the beam structure is damaged; (3) The stiffness of each position of the beam structure is obtained by dividing the load by the curvature of the influence line of the inclination angle, and the damage position is identified by the sudden change of the stiffness curve in the damage state; (4) Eliminate the stiffness at the damaged position, and fit the remaining stiffness curve to obtain the stiffness curve in the undamaged state; (5) Calculate the damage degree from the stiffness curves of the damaged and undamaged states, and obtain the structural stiffness of the damaged position. 2. The cantilever beam damage identification method based on the curvature of the free end inclination angle influence line according to claim 1, characterized in that: in step (2), the inclination angle influence line curvature θ" is calculated by the center difference: Among them, the subscript i is the number of the measuring point, θ″ _i is the curvature of the inclination influence line of the measuring point i, ε is the average value of the distance from the measuring point i-1 to the measuring point i and the distance from the measuring point i to the measuring point i+1 , θi is the inclination angle of the test position when the load acts on the _i measuring point. 3. the cantilever beam damage identification method based on free end inclination influence line curvature according to claim 2, is characterized in that: in step (3), the calculation method of structural damage state stiffness curve B _d is: Among them, B _di is the stiffness of the damage state of the i measuring point, P is the moving load value, θ″ _i is the curvature of the inclination influence line of the load acting on the i measuring point, n is the number of measuring points, and the measuring point 1 is arranged on the cantilever beam At the fixed support end, n measuring points are arranged at the free end of the cantilever beam. The number of measuring points is continuous, increasing from 1 to n, and i is greater than or equal to 2 and less than or equal to n-1. 4. The cantilever beam damage identification method based on the curvature of the free end inclination influence line according to claim 3, characterized in that: in step (3), if the sudden change of the stiffness curve is not obvious, then further seek the slope of the stiffness curve to assist judgment The calculation method of the slope of the damage position and stiffness curve is as follows: Among them, B′ _dj is the slope of the stiffness curve in the damage state of measuring point j, and ε ^* is the distance from measuring point j-1 to measuring point j. 5. the cantilever beam damage identification method based on the free end inclination angle influence line curvature according to claim 4, is characterized in that: in step (4), adopt linear fitting for the stiffness curve of equal cross-section beam undamaged state, for variable The section beam is fitted with a local parabola, and the fitted stiffness curve B _u in the undamaged state is: B _u = [0 B _u2 ... B _ui ... B _u(n-1) 0]; Among them, B _ui is the stiffness of the undamaged state fitted to the i-th measuring point. 6. the cantilever beam damage identification method based on the free end inclination angle influence line curvature according to claim 5, is characterized in that: in step (5), the calculation method of structural damage degree quantitative index _De is: D _e = [0 D _e2 ... D _ei ... D _e(n-1) 0]; Among them, _{De ei} is the structural damage degree identified by the i-th measuring point, and the calculation method is: 7. The damage identification method for cantilever beams based on the curvature of free end inclination angle influence line according to claim 1, characterized in that: in step (1), there are no less than 6 measuring points of structure inclination angle influence line.