Single damage identification method for beam structures with load-induced support rotation difference

Abstract

This invention discloses a method for identifying single-damage beam structures based on the difference in support rotation angles under load. The steps are as follows: Apply load to each span of the beam structure before and after damage; obtain the measured rotation angle values ​​of each support before and after damage using intelligent supports; for continuous beam structures, also obtain the measured support reaction force values ​​before and after damage; the measured support reaction force values ​​for two-span continuous beams can be calculated; determine the stiffness of the beam structure before damage using pre-damage information; for continuous beam structures, calculate the relative difference in measured rotation angle values ​​of each support before and after damage to determine the span where the damage occurs; finally, select one support to the left and one to the right of the damage location to form a system of equations, and then solve for the damage location and degree. This invention only requires data measured using intelligent supports, is simple to apply loads to, and is easy to operate. It can accurately locate and quantify single-damage beam structures and can be applied to damage assessment of beam structures.

Description

Technical Field

[0001] This invention relates to the field of beam structure damage detection technology, and in particular to a method for identifying single damage in beam structures based on the difference in support rotation angle under load. Background Technology

[0002] Many existing bridges no longer meet functional requirements, and safety accidents such as bridge fractures and collapses occur frequently. Civil engineering scholars have gradually recognized the importance of health monitoring and safety assessment of bridge structures and have researched various damage identification technologies. Structural damage identification is a crucial component of bridge structural health monitoring systems. Currently, there are two main categories of damage identification methods: one is based on dynamic parameters, which primarily judges structural damage by changes in structural modes (vibration frequencies and mode shapes). This type of method has high requirements for the number of measuring points, sensor measurement accuracy, and modal parameter identification methods. The other type is based on static parameters. Static parameter-based structural damage identification methods can effectively avoid the influence of uncertainties in mass, especially damping. Furthermore, due to the advanced and mature measurement equipment and technology available today, fairly accurate measurements of the structure can be obtained at a relatively low cost. Therefore, static parameter-based structural damage identification technology has received extensive research attention.

[0003] The most studied indicators for structural damage identification technology based on static parameters are deflection, static strain, and support reaction force. With the development of intelligent support technology, the change of structural rotation angle before and after damage is expected to be applied to beam structure damage identification. At present, there are few literature reports on combining support reaction force and rotation angle and applying them to the field of structural damage identification. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides a simple and low-cost method for identifying single damage in beam structures based on load-induced support rotation angle differences.

[0005] The technical solution of this invention to solve the above problems is: a method for identifying single damage in beam structures based on load-induced support rotation angle difference, characterized by comprising the following steps:

[0006] (1) Apply loads to the beam structure before and after damage, (a) for simply supported beams, obtain the measured values ​​of the support rotation angles of the two supports before and after damage; (b) for continuous beams with three or more spans, obtain the measured values ​​of the support rotation angles and support reactions of each support before and after damage; (c) for two-span continuous beams, obtain the measured values ​​of the support rotation angles of the three supports before and after damage. The measured values ​​of the support reactions can be measured or not, depending on the actual situation.

[0007] (2) The stiffness EI of the beam structure before damage is obtained by using the measured values ​​of the support rotation angle and the support reaction force before structural damage.

[0008] (3) Determine the number of spans where the damage is located. (a) For a simply supported beam, there is only one span, and the damage is located in that span. (b) For a continuous beam, calculate the relative difference between the measured values ​​of the support rotation angles before and after the structural damage to determine the number of spans where the damage is located.

[0009] (4) Identify the damage location and degree of damage of the beam structure by the measured values ​​of the support rotation angle and the support reaction force after structural damage; Assume that the damage length of the beam structure is known, the length is y, the damage location and degree of damage are unknown, the distance of the damage location from the left support is a, the stiffness at the damage location is zEI, and the degree of damage is 1-z; (a) For a simply supported beam, if the measured values ​​of the two support rotation angles after structural damage are equal to the calculated values, list the two support rotation angle equations, form a system of equations, and solve for the damage location a and the damage stiffness zEI;

[0010] (b) For continuous beams, select the measured values ​​of the support rotation angles of one support on the left and one on the right of the span where the damage occurs, and take the basic structure of a simply supported beam. For two spans of continuous beams where the support reaction force is not measured, calculate the support reaction force value using the force method or displacement method. For other cases of continuous beams, use the measured value of the support reaction force. Replace the support with the support reaction force value to obtain the calculated value of the support rotation angle. Set up two sets of equations based on the equality of the measured value and the calculated value of the support rotation angle, and solve for the damage location a and the damage stiffness zEI.

[0011] Specifically, in step (1), the measured vectors of support rotation angles and support reaction forces before and after structural damage are obtained as needed:

[0012] ;

[0013] ;

[0014] ;

[0015] ;

[0016] in, , These are the measured rotation angles of support j before and after structural damage in the i-th span of the n-span beam, respectively. The superscript n indicates the number of spans of the beam, and the subscript... This indicates that the load is applied to the i-th span, the subscript j indicates support j, u indicates before structural damage, and d indicates after structural damage. , These are the measured reaction forces at support j before and after structural damage in the i-th span when the load of the n-span beam is applied.

[0017] Specifically, in step (2), the stiffness EI before structural damage is obtained:

[0018] We can obtain the following by integration:

[0019] ;

[0020] in, The calculated value of the rotation angle of support j before the damage of the i-th span structure under the load of the n-span beam. This is the total length of the beam structure; Apply a unit moment of bending at the j-th support of the n-span beam; Let be the bending moment when an n-span beam load is applied to the i-th span.

[0021] Substituting the measured values ​​of the rotation angle, the stiffness EI before structural damage can be obtained;

[0022] .

[0023] Specifically, in step (3), the relative difference between the measured values ​​of the rotation angle before and after the support damage is:

[0024] ;

[0025] When the relative differences between the measured rotation angles of adjacent supports are equal, the damage is not located in this span, that is, when If the damage is not in the j-th span, the number of spans where the damage occurs is determined by considering the load acting on each span.

[0026] Specifically, in step (4), we can obtain two equations about the rotation angle:

[0027] For simply supported beams:

[0028] ;

[0029] ;

[0030] For a two-span continuous beam, when the measured values ​​of the support reactions are obtained:

[0031] ;

[0032] In this case, it is assumed that the damage is located in the first span and the load is applied to the second span. Let be the bending moment of the simply supported beam basic structure corresponding to the i-th span of the n-span beam; This is the calculated value of the reaction force at support j when the load is applied to the i-th span of the n-span beam. Let be the bending moment of the simply supported beam basic structure corresponding to the j-th support of the n-span beam under a unit vertical load.

[0033] The solution can be obtained by substituting the measured values ​​of the support reactions into the formula:

[0034] ;

[0035] ;

[0036] For a two-span continuous beam, without measuring the actual support reactions:

[0037] First, solve for X using the force method equations; the other steps are the same as above.

[0038] ;

[0039] ;

[0040] ;

[0041] ;

[0042] in, The displacement along the X direction produced by a unit force X is called the compliance coefficient; X is the unit force. The displacement along the X direction is caused by the load;

[0043] For continuous beams with three or more spans:

[0044] ; ;

[0045] It is assumed that the damage has been determined to be located in the m-th span through step (3), and the solution is performed using support m and support m+1;

[0046] The system of equations has three unknowns: a, y, and z. EI has been obtained through step (2), and y is assumed to be known. Therefore, with two equations and two unknowns, a and z can be solved.

[0047] Specifically, in step (4), if the beam structure cross section is a variable cross section, then the moment of inertia is a variable, and the method still applies.

[0048] Specifically, if the damage location is at the inflection point, i.e. at the zero bending moment, the solution accuracy may be poor. Changing the load condition so that the damage location is not near the inflection point can result in an accurate solution.

[0049] Specifically, in step (4), the damage length y of the beam structure can be taken as 1 / 50 to 1 / 4 of the damage span; when solving the equation system, the initial value of the damage location a is selected after determining the number of spans where the damage is located, and is generally taken as the starting position of the span up to the middle span.

[0050] Specifically, in step (4), the basic structure can also be a cantilever beam structure.

[0051] Specifically, in step (4), if the damage is located in the m-th span, selecting the support rotation angles of m and m+1 to form a system of equations will improve noise resistance.

[0052] The beneficial effects of this invention are as follows: This invention applies loads to the beam structure before and after damage, obtaining measured values ​​of the support rotation angles at each support under the corresponding conditions. For continuous beams, measured values ​​of support reactions before and after damage are also required; the support reactions of two-span continuous beams can also be calculated. The bending stiffness of the beam structure is obtained using information prior to damage. Identification of continuous beams also requires determining the span number of the damaged beam by the relative difference between the measured support rotation angles before and after structural damage. Simultaneously, a system of equations consisting of the rotation angles of one support on each side of the structural damage location is established, and the damage location and degree are solved using this rotation angle equation system. A solution approach for n-span continuous beams is summarized and proposed. Through examples of simply supported beams, two-span unequal-span continuous beams, and five-span continuous beams, the application value of the rotation angle difference index in single-damage identification of beam structures is verified, providing an effective new method for the location and quantification of beam structure damage. Attached Figure Description

[0053] Figure 1 This is a flowchart of the method of the present invention.

[0054] Figure 2 This is a model diagram of the simply supported beam structure of the present invention.

[0055] Figure 3 This is the bending moment diagram of a simply supported beam subjected to a concentrated load at mid-span, according to the present invention.

[0056] Figure 4 This is the bending moment diagram of support 1 for a simply supported beam under unit couple action according to the present invention.

[0057] Figure 5 This is the bending moment diagram of the support 2 of the simply supported beam under the action of a unit couple according to the present invention.

[0058] Figure 6 This is a diagram of the equivalent linear stiffness model of the damage at the measuring point in this invention.

[0059] Figure 7 This is a model diagram of the two-span continuous beam of the present invention.

[0060] Figure 8 This is the bending moment diagram of the basic structure of the two-span continuous beam of the present invention under unit force acting on support 2.

[0061] Figure 9 This is the bending moment diagram of the basic structure of the simply supported beam corresponding to the mid-span of the right span under the concentrated load of the two-span continuous beam of the present invention.

[0062] Figure 10 This is the bending moment diagram of the unit force couple acting on support 1 of the two-span continuous beam of the present invention.

[0063] Figure 11 This is the bending moment diagram of the unit force couple acting on support 2 of the two-span continuous beam of the present invention.

[0064] Figure 12This is the bending moment diagram of the unit force couple acting on support 3 of the two-span continuous beam of the present invention.

[0065] Figure 13 This is a model diagram of the five-span continuous beam of the present invention.

[0066] Figure 14 This is a cross-sectional view of the five-span continuous beam T-beam of the present invention. Detailed Implementation

[0067] The present invention will be further described below with reference to the accompanying drawings and embodiments. When the following description refers to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements.

[0068] like Figure 1 As shown, a method for identifying single damage in a beam structure based on the support rotation difference under load is described below. The specific steps are as follows:

[0069] Step 1: Apply loads to the beam structure before and after damage, span by span. (a) For simply supported beams, obtain the measured values ​​of the support rotation angles of the two supports before and after damage. (b) For continuous beams with three or more spans, obtain the measured values ​​of the support rotation angles and support reactions of each support before and after damage. (c) For two-span continuous beams, obtain the measured values ​​of the support rotation angles of the three supports before and after damage. The measured values ​​of the support reactions can be measured or not, depending on the actual situation.

[0070] Step 2: Calculate the stiffness EI of the beam structure before damage by using the measured values ​​of the support rotation angle and support reaction force before structural damage.

[0071] Step 3: Determine the span in which the damage is located. (a) For a simply supported beam, there is only one span, and the damage is located in that span. (b) For a continuous beam, calculate the relative difference between the measured support rotation angles before and after the structural damage to determine the span in which the damage is located.

[0072] Step 4: Identify the damage location and degree of the beam structure by using the measured values ​​of the support rotation angle and support reaction force after structural damage; Assume that the damage length of the beam structure is known, the length is y, the damage location and degree of damage are unknown, the distance of the damage location from the left end support is a, the stiffness at the damage location is zEI, and the degree of damage is 1-z; (a) For a simply supported beam, if the measured values ​​of the two support rotation angles after structural damage are equal to the calculated values, list the two support rotation angle equations, form a system of equations, and solve for the damage location a and the damage stiffness zEI;

[0073] (b) For continuous beams, select the measured values ​​of the support rotation angles of one support on the left and one on the right of the span where the damage occurs, and take the basic structure of a simply supported beam. For two spans of continuous beams where the support reaction force is not measured, calculate the support reaction force value using the force method or displacement method. For other cases of continuous beams, use the measured value of the support reaction force. Replace the support with the support reaction force value to obtain the calculated value of the support rotation angle. Set up two sets of equations based on the equality of the measured value and the calculated value of the support rotation angle, and solve for the damage location a and the damage stiffness zEI.

[0074] (I) Derivation of the Theory of Simply Supported Beams

[0075] Structural model such as Figure 2 As shown, the span is L. Assume that the damage is located in the left half-span and the distance from the damage location to the leftmost support is a. The same applies when it is located in the right half-span. The length of the damage is y. The structural stiffness at the undamaged location is EI, and the structural stiffness at the damaged location is zEI (0 < z < 1).

[0076] Apply step 1 to obtain the measured values ​​of the rotation angles of each support before and after damage to the simply supported beam structure.

[0077] (1)

[0078] (2)

[0079] in, , These are the measured rotation angles of support j before and after structural damage in the i-th span of the n-span beam, respectively. The superscript n indicates the number of spans of the beam, and the subscript... This indicates that the load is applied to the i-th span, the subscript j indicates support j, u indicates before structural damage, and d indicates after structural damage.

[0080] Applying step 2, the concentrated load F acts on the mid-span bending moment as (e.g.) Figure 3 ):

[0081] (3)

[0082] in, The superscript 1 indicates a single-span beam; the first subscript 1 indicates that the concentrated load is applied to the first span, and the second subscript 1 indicates the first part.

[0083] The bending moment when a unit couple is applied to support 1 is (e.g.) Figure 4 ):

[0084] (4)

[0085] in, The superscript 1 indicates a single-span beam; the first subscript 1 indicates that a unit couple is acting on support 1; the second subscript 1 indicates the first part.

[0086] The bending moment when a unit couple is applied to support 2 is (e.g.) Figure 5 ):

[0087] (5)

[0088] The expression for the angle is:

[0089] (6)

[0090] Where n represents the number of spans in the beam structure, i represents the location of the load application, and j represents the support number. This represents the actual bending moment of the beam structure. This represents the structural bending moment of a unit moment acting on support j. The actual bending moment of a simply supported beam structure is the same as the bending moment under concentrated load.

[0091] In summary, the expression for the concentrated load F acting on the first span support 1 and support 2 is as follows:

[0092] (7)

[0093] (8)

[0094] EI is obtained from the rotation angle information before the damage, here it is obtained through the rotation angle of support 1:

[0095] (9)

[0096] (10)

[0097] By substituting the measured data, the EI can be calculated.

[0098] Applying step 3, for simply supported beams, there is no need to determine the number of damaged spans; there is only one span, and the damage is located in that span.

[0099] Applying step 4, the rotation angles of the two supports under concentrated load are obtained, resulting in two equations. However, there are currently three unknowns: a, y, and z. With three unknowns and two equations, and assuming y is known, the two equations and two unknowns can be solved using mathematical software. By aligning the calculated values ​​with the measured values ​​and rearranging EI, the solution can be obtained.

[0100] (11)

[0101] (12)

[0102] In practical engineering, the damage length is also unknown. Analyzing the equivalent damage degree, the damage interval divides the measuring point interval into several parts with different damage degrees, which can be regarded as several segments with different linear stiffness damage degrees connected in series. For example... Figure 6As shown: Analyze the damage degree of the beam segment between measuring points 2 and 3. The stiffness of the undamaged part is... , length is The shaded area represents the stiffness of the damaged part. , length is .

[0103] By the principle of series connection of spring stiffness, if n springs are connected in series, then the reciprocal of the total stiffness is equal to the sum of the reciprocals of the individual stiffnesses, that is:

[0104] (13)

[0105] The equivalent stiffness of the two measuring points before damage is:

[0106] (14)

[0107] In the formula:

[0108] (15)

[0109] (16)

[0110] (II) Theoretical Derivation of Two-Span Unequal-Span Continuous Beams

[0111] Two-span unequal span continuous beam structure, such as Figure 7 As shown, first assume that force F acts on the undamaged span, and then a concentrated force F is applied at the mid-span of the right span. The distance from the damaged location to the leftmost support is 'a', the damage length is 'y', and the spans of the two spans are respectively... The lengths are respectively and The stiffness of the undamaged part is EI, and the stiffness of the locally damaged element is zEI.

[0112] Applying step 1, as needed, obtain the measured values ​​of the rotation angles of each support before and after damage to the two-span unequal span continuous beam structure.

[0113] (17)

[0114] (18)

[0115] The basic structural bending moment diagram of the unit force acting on support 2 is as follows: Figure 8 As shown, the bending moment diagram of the simply supported beam basic structure corresponding to the mid-span of the right span under concentrated load is as follows. Figure 9 As shown, the bending moment diagrams for applying a unit moment to supports 1-3 are as follows. Figures 10-12 As shown.

[0116] The above diagram shows the bending moment of the basic structure under a unit force. The expression is:

[0117] (19)

[0118] The basic bending moment expression for a simply supported beam subjected to a concentrated load F at the mid-span of the right span is:

[0119] (20)

[0120] The expressions for the bending moments obtained by applying a unit couple to supports 1-3 are as follows:

[0121] (twenty one)

[0122] (twenty two)

[0123] (twenty three)

[0124] When the beam is damaged, we can obtain the following using the graphical multiplication method:

[0125] (twenty four)

[0126] (25)

[0127] (26)

[0128] (27)

[0129] in, The displacement along the X direction produced by a unit force X is called the compliance coefficient; X is the unit force. The displacement along the X direction is caused by the load; Let be the bending moment of the simply supported beam basic structure corresponding to the i-th span of the n-span beam; Let be the bending moment of the simply supported beam basic structure corresponding to the j-th support of the n-span beam under a unit vertical load.

[0130] Apply step 2 to obtain the EI using the pre-damage information.

[0131] (28)

[0132] (29)

[0133] Substituting the measured data into the equations, the EI before structural damage can be calculated.

[0134] Applying step 3, by comparing the relative difference between the measured values ​​of the support reaction force before and after the damage, if two adjacent values ​​are equal, then the damage is not located in this span.

[0135] (30)

[0136] Applying step 4, the corner expression is listed below:

[0137] (31)

[0138] (32)

[0139] (33)

[0140] From the above, we can obtain three rotation angle expressions. Then, by measuring the rotation angles of the three supports in actual engineering, and setting the calculated values ​​equal to the theoretical values, and rearranging EI terms, we get:

[0141] (34)

[0142] (35)

[0143] When using step 3, an index of the relative difference before and after support rotation damage is proposed to distinguish which span the damage is located in, so as to facilitate further analysis. The proof process is given here.

[0144] Here, we need to discuss two cases: the concentrated load F is applied in the middle of the damaged span and the undamaged span. Let's first analyze the case where the load is applied in the middle of the undamaged span.

[0145] (36)

[0146] (37)

[0147] (38)

[0148] The expression for the relative difference in support rotation angle is:

[0149] (39)

[0150] in, The superscript 2 indicates the span number, the subscript i indicates that the load is applied to the i-th span, and j indicates the support j;

[0151] (40)

[0152] (41)

[0153] (42)

[0154] We now continue our analysis of the concentrated load F acting at the mid-span of the damaged span, i.e., at the mid-span of the first span. The analysis process is the same as when the load acts on the undamaged span. The rotation angles of the three supports after the damage are:

[0155] (43)

[0156] (44)

[0157] (45)

[0158] (46)

[0159] The rotation angles of the three supports before the damage were:

[0160] (47)

[0161] in, This is the calculated value of the reaction force before damage to support 2 when the load is applied to the first span of the two-span beam. The superscript 2 indicates the number of spans, the subscript 1 indicates that the load is applied to the first span, u indicates before damage, and 2 indicates support 2.

[0162] (48)

[0163] (49)

[0164] (50)

[0165] (51)

[0166] (52)

[0167] (53)

[0168] It can be seen that the relative rotation angle difference between support 2 and support 3 is exactly equal, which indicates that the damage is not located in the second span, but is located in the first span, that is, between support 1 and support 2.

[0169] (III) Theoretical Derivation of n-Span Continuous Beams

[0170] The theoretical derivation of an n-span continuous beam is now performed. Applying step 1, the measured vectors of support rotation angles and support reactions before and after structural damage are obtained as needed:

[0171] (54)

[0172] (55)

[0173] (56)

[0174] (57)

[0175] Apply step 2 and substitute the measured data to solve for the EI of the beam structure before damage.

[0176] (58)

[0177] (59)

[0178] Applying step 3, the relative difference between the measured values ​​before and after the support rotation angle damage is compared to identify which span the damage is located in.

[0179] (60)

[0180] Applying step 4, randomly select one support rotation angle on the left and right sides of the damage, derive its expression, substitute it into the data, and list the rotation angle equations to form a system of equations.

[0181] (61)

[0182] (62)

[0183] In this case, the damage is assumed to be located in span m, and the solution is performed using supports m and m+1.

[0184] Example 1: Simply supported beam

[0185] Consider a simply supported beam bridge with a span of 100cm. Divide the structure into 20 elements (5cm intervals each) and 21 nodes. The cross-section is 5cm wide and 3cm high. E= Pa. Now assume the damage location is 30cm from the support, the damage degree is 0.3, and the damage length is 5cm.

[0186] Use step 1 to obtain information about the beam structure before and after damage.

[0187] With a concentrated load of F=50N, the theoretical rotation angle of support 1 is 0.0106746032. The simulated value through ANSYS is -0.0106746032. When 10 decimal places are retained, the simulated value is equal to the actual value. The reason for the different directions is that this article specifies that the bending moment is positive when it is at the top, while ANSYS defaults to the bending moment being positive when it is at the bottom.

[0188] With a concentrated load of F=50N, the theoretical rotation angle of support 2 is -0.0104747208. The simulated value through ANSYS is 0.0104747208. After retaining 10 decimal places, the simulated value is equal to the actual value.

[0189] Using step 3, substitute the data to obtain the EI.

[0190] (63)

[0191] Step 3 is not required for simply supported beams.

[0192] Using step 4, since the angle expression contains a large term EI, we consider optimizing the equation system by multiplying each term by EI and simplifying EI to a constant term, which can improve the accuracy of the equation system. The following two equations about the angle can be listed, forming a nonlinear equation system. Because this paper selects the bending moment at the top as positive, the calculated value and the measured value are opposites and add up to zero.

[0193] The system of equations will now be solved using mathematical software, specifically a nonlinear equation system solver. Since it is a nonlinear system of equations, an iterative method is required, necessitating the provision of an initial value.

[0194] (64)

[0195] A loop is written to increment the initial value of 'a' in the equation system from 0.05 to 0.5, with each increment representing the length of the damage. The initial value of 'z' is arbitrary; we assume it to be 0.8. First, we solve the case where 'y' is 5 cm. The results are shown in Table 1. It can be seen that, by retaining four decimal places, this method can accurately locate and quantify the damage, yielding the true value. Changing the initial value of 'z' to 0.3 does not change the result; therefore, in this method, the given initial value 'z' has a relatively small impact on the solution.

[0196] Table 1. Damage identification results of simply supported beams (damage length of beam segment is 0.05m)

[0197]

[0198] The solution above assumed the actual damage length. However, in actual engineering, the damage length is unknown. To further demonstrate the practicality of this method, and considering that the damage is relatively small, we now assume the damage length y is 10cm, keeping other parameters unchanged. The results are shown in Table 2:

[0199] Table 2 Damage identification results of simply supported beams (damage length of beam segment is 0.1m)

[0200]

[0201] The obtained damage location is (0.2731, 0.3731), which includes the true value (0.3, 0.35) and can perform relatively accurate damage identification. In actual engineering, after determining the approximate location, it can be further located more precisely with the help of detection instruments. The obtained z is 0.8227.

[0202] (65)

[0203] Assuming a damage length y of 10cm, the calculated stiffness can also be used to calculate the actual stiffness through a series spring model with high accuracy. The difference between the solution and the calculated value is only 0.0008. Due to the influence of measurement accuracy, there is a small difference after retaining 4 decimal places, which can be considered as the true value. Therefore, the single-damage identification method for beam structures based on the support rotation difference under load has high accuracy in damage location and quantification.

[0204] Example 2: Two-span unequal span continuous beam

[0205] Consider a continuous beam bridge with a span of 200cm, where the two spans are 80cm and 120cm long respectively. Divide the structure into 5cm elements, resulting in 40 elements and 41 nodes. The cross-section is 5cm wide and 3cm high. E = Pa. Now assume that the location of the damage is 30cm from support 1, the degree of damage is 0.3, and the length of the damage is 5cm.

[0206] Use step 1 to obtain beam structure information.

[0207] The theoretical rotation angle of support 1 under a concentrated load of F=50N is -0.0030613981, and the simulated value through ANSYS is 0.0030613981. With 10 decimal places retained, the simulated value is equal to the actual value.

[0208] The theoretical rotation angle of support 2 under a concentrated load of F=50N is 0.0059729231, while the simulated value through ANSYS is -0.0059729231. With 10 decimal places retained, the simulated value is equal to the actual value.

[0209] The theoretical rotation angle of support 3 under a concentrated load of F=50N is -0.0103938690°, and the simulated value through ANSYS is 0.0103938690°. With 10 decimal places retained, the simulated value is equal to the actual value.

[0210] Using step 2, substitute the data to obtain the EI.

[0211] (66)

[0212] Using step 3, calculate the relative difference between the measured rotation angles of each support before and after damage. The results are shown in Table 3.

[0213] Table 3. Relative differences of measured rotation angles of two-span continuous beams

[0214]

[0215] It can be seen that when the load is applied to the first span, the relative difference between the measured values ​​of the rotation angles of support 2 and support 3 is equal, therefore the damage is not located in the second span, but in the first span.

[0216] Using step 4, solve the angle equations. Consider optimizing the system of equations by multiplying each equation by EI and adjusting the position of EI into the numerical terms to reduce the difference between the two terms and improve accuracy.

[0217] (67)

[0218] (68)

[0219] (69)

[0220] We obtained three angle equations, and three unknowns. Since y is a known quantity that we assume, theoretically, we only need two of the three equations to solve them. Choosing two of the three equations results in three possible combinations, which we will analyze one by one. The solutions to equations 1 and 2 are shown in Table 4, the solutions to equations 1 and 3 are shown in Table 5, and the solutions to equations 2 and 3 are shown in Table 6.

[0221] Table 4 Equations 1 and 2 for a two-span beam (damage length of beam segment 0.05m)

[0222]

[0223] Table 5 Equations 1 and 3 for two-span beams (damage length of beam segment 0.05m)

[0224]

[0225] Table 6 Equations 2 and 3 for two-span beams (damage length of beam segment 0.05m)

[0226]

[0227] It is evident that the solution using supports 1 and 2 has high accuracy, with a and z being the true values. The solution using supports 1 and 3 also has high accuracy, with only a few instances where a differs by as little as 0.0001, and z being the true value. However, using supports 2 and 3 reveals that the solution is completely unsolvable. The reason for this is that the damage is located between supports 1 and 2. The simplified equations for supports 2 and 3 are highly similar, differing by only one term. From a mathematical perspective, we can consider the two equations to be related. Therefore, when solving two related equations, there are equivalent cases, resulting in multiple solutions. As can be seen from the above... and Only and The difference is that the other terms are the same, so it is impossible to solve.

[0228] The above text derives the relative difference before and after support damage. To make it more intuitive, ANSYS is used to calculate the relative difference of support rotation angle when the load is applied to the undamaged span and the damaged span, respectively.

[0229] Based on the combined expressions and example data, it can be concluded that when the load is applied to the undamaged span, only the equations formed using supports 2 and 3 cannot be solved; the other two sets have high solution accuracy. However, when a concentrated load is applied to the damaged span, the relative differences in the rotation angles of the two supports in the undamaged span before and after damage are equal, indicating that the damage is not in this span, but in another span. By applying two concentrated loads, the span in which the damage is located can be distinguished. Applying concentrated loads is relatively easy to implement in engineering, and the rotation angle can be directly read from the intelligent supports.

[0230] This method allows for accurate damage identification by measuring only the bridge's support rotation angle and applying two concentrated loads, which is convenient in engineering and has broad application prospects.

[0231] Example 3: Five-span continuous beam

[0232] Analyzing a five-span continuous beam bridge as a representative of an n-span beam, such as... Figure 13 As shown.

[0233] Each span is 20m, with a T-beam cross-section, using C50 concrete, and E is... The surface is divided into elements of 0.5m, resulting in a total of 200 elements and 201 nodes. The cross-sectional shape is as follows: Figure 14 As shown.

[0234] Assume damage occurs in the third span, with damage location 'a' 45m from the leftmost support 1, and damage severity 0.5. In practical engineering, applying a single concentrated load is difficult; therefore, a two-axle vehicle is used to load the beam structure. The vehicle has a total weight of 200kN, with 80kN on the front axle and 120kN on the rear axle, and a wheelbase of 5m. The vehicle is driven to a position where the front axle is 50m from support 1 and the rear axle is 55m from support 1.

[0235] Apply step 1 to obtain the information needed to solve the problem.

[0236] Apply step 2 and substitute the data to solve for the EI before damage.

[0237] (70)

[0238] Applying step 3, the number of spans where the damage occurred was identified by comparing the relative differences in the rotation angles before and after the damage. The two-axle vehicle was driven to a position where the front axle was in the middle of the first span and the rear axle was 5m to the right of the front axle. This process was then repeated in other spans. The relative differences in the measured rotation angles before and after the damage are shown in Table 7.

[0239] Table 7. Measured values ​​of rotation angle of a five-span continuous beam before and after damage; relative differences.

[0240]

[0241] When the load is applied to the first span, it can be determined that the damage is not located in the fourth or fifth span; when the load is applied to the second span, it can be determined that the damage is not located in the first, second, fourth, or fifth span; when the load is applied to the third span, it can be determined that the damage is not located in the first, second, fourth, or fifth span; when the load is applied to the fourth span, it can be determined that the damage is not located in the first, second, or fifth span; when the load is applied to the fifth span, it can be determined that the damage is not located in the first or second span; therefore, the damage is located in the third span.

[0242] Applying step 4, list the rotation angle equation and solve it using the rotation angles of the two supports 3 and 4 on the left and right sides of the damage span. The solution results are shown in Table 8.

[0243] The measured rotation angles of supports 3 and 4 are 0.0001697984 and -0.0002277763, respectively.

[0244] Table 8 Damage identification results of a five-span beam (damage length of beam segment is 0.05m)

[0245]

[0246] It can be seen that the solution accuracy is very high. Therefore, this theory is also applicable to five-span continuous beams and can be applied to continuous beams with any number of spans. It has a broad prospect of application in practical engineering.

[0247] The above descriptions are merely three embodiments of the present invention. All equivalent changes and modifications made within the scope of the claims of the present invention are within the scope of the present invention.

Claims

1. A method for identifying single damage in beam structures based on support rotation difference under load, characterized in that, Includes the following steps: (1) Apply loads to the beam structure before and after damage, (a) for simply supported beams, obtain the measured values ​​of the support rotation angles of the two supports before and after damage; (b) for continuous beams with more than three spans, obtain the measured values ​​of the support rotation angles and support reactions of each support before and after damage; (c) for two-span continuous beams, obtain the measured values ​​of the support rotation angles of the three supports before and after damage. (2) The stiffness EI of the beam structure before damage is obtained by using the measured values ​​of the support rotation angle and the support reaction force before structural damage. (3) Determine the number of spans where the damage is located. (a) For a simply supported beam, there is only one span, and the damage is located in that span. (b) For a continuous beam, calculate the relative difference between the measured values ​​of the support rotation angles before and after the structural damage to determine the number of spans where the damage is located. (4) Identify the damage location and degree of damage of the beam structure by the measured values ​​of the support rotation angle and the support reaction force after structural damage; Assume that the damage length of the beam structure is known, the length is y, the damage location and degree of damage are unknown, the distance from the damage location to the left support is a, the stiffness at the damage location is zEI, and the degree of damage is 1-z; (a) For a simply supported beam, if the measured values ​​of the two support rotation angles after structural damage are equal to the calculated values, list the two support rotation angle equations, form a system of equations, and solve for the damage location a and the damage stiffness zEI; (b) For continuous beams, select the measured values ​​of the support rotation angles of one support on the left and one on the right of the span where the damage occurs, and take the basic structure of a simply supported beam. For two spans of continuous beams where the support reaction force has not been measured, calculate the support reaction force value using the force method or displacement method. For other continuous beams where the support reaction force has been measured, use the measured support reaction force value to replace the support and obtain the calculated support rotation angle value. Since the measured and calculated support rotation angle values ​​are equal, set up two sets of equations to solve for the damage location a and the damage stiffness zEI. In step (1), the measured vectors of support rotation angles and support reaction forces before and after structural damage are obtained as needed: ； ； ； ； in, , These are the measured rotation angles of support j before and after structural damage in the i-th span of the n-span beam, respectively. The superscript n indicates the number of spans of the beam, and the subscript... This indicates that the load is applied to the i-th span, the subscript j indicates support j, u indicates before structural damage, and d indicates after structural damage. , These are the measured values ​​of the reaction forces at support j before and after the damage to the i-th span structure under the load of the n-span beam. In step (2), the stiffness EI of the beam structure before damage is obtained: We obtain this by integration: ； in, The calculated value of the rotation angle of support j before the damage of the i-th span structure under the load of the n-span beam. This is the total length of the beam structure; Apply a unit moment of bending at the j-th support of the n-span beam; Let be the bending moment when an n-span beam load is applied to the i-th span. Substitute the measured values ​​of the rotation angle to obtain the stiffness EI of the beam structure before damage. ； In step (3), the relative difference between the measured values ​​of the rotation angle before and after the support damage is: ； When the relative differences in the measured rotation angles before and after damage to adjacent supports are equal, the damage is not located in this span, that is, when When the damage is not in the j-th span, the number of spans where the damage is located is determined by considering the load acting on each span. In step (4), two equations about the support rotation angle are obtained: For simply supported beams: ； ； For a two-span continuous beam, when the measured values ​​of the support reactions are obtained: ； In this case, it is assumed that the damage is located in the first span and the load is applied to the second span. Let be the bending moment of the simply supported beam basic structure corresponding to the i-th span of the n-span beam; This is the calculated value of the reaction force at support j when the load is applied to the i-th span of the n-span beam. Let be the bending moment of the simply supported beam basic structure corresponding to the j-th support of the n-span beam under a unit vertical load. The solution is obtained by substituting the measured values ​​of the support reactions into the formula: ； ； For a two-span continuous beam, without measuring the actual support reactions: First, solve for X using the force method equations; the other steps are the same as above. ； ； ； ； in, The displacement along the X direction produced by a unit force X is called the compliance coefficient; X is the unit force. The displacement along the X direction is caused by the load; For continuous beams with three or more spans: ； ； It is assumed that the damage has been determined to be located in the m-th span through step (3), and the solution is performed using support m and support m+1; The system of equations has three unknowns: a, y, and z. EI has been obtained through step (2), and y is assumed to be known. Therefore, there are two unknowns in the two equations. Solve for a and z.

2. The method for identifying single damage in beam structures based on load-induced support rotation angle difference according to claim 1, characterized in that: In step (4), if the beam structure cross section is a variable cross section, then the moment of inertia is a variable.

3. The method for identifying single damage in beam structures based on load-induced support rotation angle difference according to claim 1, characterized in that: In step (4), the damage length y of the beam structure is taken as 1 / 50 to 1 / 4 of the span of the damaged span; when solving the equation system, the initial value of the damage location a is selected after determining the number of spans where the damage is located, and the starting position of the span is taken up to no more than the middle span.

4. The method for identifying single damage in beam structures based on load-induced support rotation angle difference according to claim 1, characterized in that: In step (4), the beam structure is a cantilever beam structure.

5. The method for identifying single damage in beam structures based on load-induced support rotation angle difference according to claim 1, characterized in that: In step (4), if the damage is located in the m-th span, select the support rotation angles of m and m+1 to set up a system of equations.

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