Single Damage Identification Method for Beam Structures with Different Deflections under Load

Abstract

This invention discloses a method for identifying single-damage beam structures based on deflection differences under load. The steps are as follows: Apply load to each span of the beam structure before and after damage, obtaining measured deflection values ​​at the mid-span and any other point within each span before and after damage. For continuous beam structures, measured support reactions before and after damage are also required; the measured support reactions for two-span continuous beams can be calculated. The flexural stiffness of the beam structure before damage is determined using the obtained pre-damage information. For continuous beam structures, the relative difference between the measured deflection values ​​at two measuring points in each span before and after damage is calculated to determine the number of spans where the damage occurs. Finally, a system of equations is formed by selecting one measuring point to the left and one to the right of the damage location, and then solving for the damage location and degree. This invention requires fewer measuring points, only requires the application of concentrated loads, and is simple and convenient 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 with load-induced deflection differences. Background Technology

[0002] In recent years, the number of old bridges in my country has been increasing, and the problems they present have become increasingly prominent. Many existing bridges can no longer meet functional requirements, and safety accidents such as bridge fractures and collapses occur frequently. 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 component of bridge structural health monitoring systems. Currently, there are two main categories of damage identification methods: one is based on dynamic parameters, which mainly 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.

[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 changes in support reaction force and deflection before and after damage are expected to be applied to beam structure damage identification. Currently, there are few literature reports on combining support reaction force and deflection 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 beam structures with poor deflection under load.

[0005] The technical solution of this invention to solve the above problems is: a method for identifying single damage in beam structures with different deflections under load, characterized by comprising the following steps:

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

[0007] (2) The stiffness EI of the beam structure before damage is obtained by measuring the deflection at the measuring points and the support reaction 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 in deflection by measuring the deflection values ​​at the measuring points before and after the structural damage, and 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 measuring the actual deflection value of the measuring point after structural damage and the actual reaction value of the support; 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 point is zEI, and the degree of damage is 1-z; (a) For a simply supported beam, if the measured deflection value of the two measuring points after structural damage is equal to the calculated value, list the deflection equations of the two measuring points, form a set of equations, and solve for the damage location a and the damage stiffness zEI;

[0010] (b) For continuous beams, select one measuring point 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 support reaction force value to replace the support and obtain the calculated value of the measuring point deflection. Since the measured value of the deflection is equal to the calculated value, set up two sets of equations to solve for the damage location a and the damage stiffness zEI.

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

[0012] ;

[0013] ;

[0014] ;

[0015] ;

[0016] in, , These are the measured values ​​of deflection at point m before and after damage to the i-th span of the beam under a concentrated load applied to the n-span beam. The superscript n indicates the number of spans of the beam, and the subscript... This indicates that a concentrated load is applied to the i-th span, the subscript m indicates the measuring point m, u indicates before structural damage, and d indicates after structural damage. , These are the measured reactions at support j before and after damage to the i-th span structure under concentrated load on the n-span beam, respectively, with the subscript j indicating the support number.

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

[0018] From the deflection expression, we can obtain:

[0019] ;

[0020] in, The calculated value of the deflection at measuring point m before damage occurs when a concentrated load is applied to the i-th span of the beam. This is the total length of the beam structure; The bending moment obtained after applying a unit force at the m-th measuring point of an 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 deflection data, the stiffness EI before structural damage can be obtained;

[0022] .

[0023] Specifically, in step (3), the relative difference between the measured values ​​of deflection before and after the damage at the measuring point is:

[0024] ;

[0025] When the relative difference in deflection between the two measuring points in the span is equal, the damage is not located in this span, that is, when and When the values ​​are equal, the damage is not in the i-th span. Considering the load acting on each span, the number of spans where the damage occurs is determined comprehensively.

[0026] Specifically, in step (4), two equations about deflection can be obtained:

[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 concentrated load is applied to the second span. Measuring point 1 is the measuring point on the left side of the first span, and measuring point 3 is the measuring point in the middle of the second span. The two measuring points are located on the left and right sides of the damage, respectively. Let be the bending moment of the simply supported beam basic system 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 rest is 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] ;

[0046] Assuming that the damage has been determined to be located in the o-th span through step (3), the mid-span measuring points of the o-1 and o+1 spans are selected to write the equation. Then, measuring point 2o-3 is the mid-span measuring point of the o-1 span, and measuring point 2o+1 is the mid-span measuring point of the o+1 span.

[0047] 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, the two unknowns of the two equations can be solved to obtain a and z.

[0048] 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.

[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. Generally, the starting position of the span is taken until the equation system has a continuous stable solution.

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

[0051] Specifically, in step (4), if the load is applied once, the equations are solved using the deflection of two measuring points, and the two measuring points must be located on both sides of the damage; if the load is applied twice, the equations are solved using the deflection of one measuring point under two working conditions, and the two loading positions must be located on both sides of the damage.

[0052] Specifically, in step (4), it is difficult to apply concentrated loads or arrange measuring points in the area near the edge support, resulting in poor identification of this area in actual engineering.

[0053] The beneficial effects of this invention are as follows: This invention applies loads to the beam structure before and after damage, obtaining the measured deflection values ​​at each measuring point of the beam structure under the corresponding conditions. For continuous beams, the measured 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 through information before damage. The identification of continuous beams also requires determining the number of spans where the damage occurs by using the relative difference between the measured deflection values ​​before and after structural damage. Simultaneously, a set of equations consisting of one deflection on each side of the structural damage location is established, and the damage location and degree are solved from the deflection equations. 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 deflection 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

[0054] Figure 1 This is a flowchart of the method of the present invention.

[0055] Figure 2 This is a model diagram of the simply supported beam structure of the present invention.

[0056] 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.

[0057] Figure 4 This is the bending moment diagram of measuring point 1 under unit force on a simply supported beam according to the present invention.

[0058] Figure 5 This is the bending moment diagram of measuring point 2 under unit force on a simply supported beam according to the present invention.

[0059] Figure 6 This is a diagram of the equivalent linear stiffness model of the damage at the measuring point in this invention.

[0060] Figure 7 This is a model diagram of the two-span continuous beam of the present invention.

[0061] Figure 8This 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.

[0062] Figure 9 This is the bending moment diagram of the basic structure of the simply supported beam corresponding to the mid-span of the left span of the two-span continuous beam under concentrated load according to the present invention.

[0063] Figure 10 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.

[0064] Figure 11 This is the bending moment diagram of a unit force acting on a measuring point 1 in a two-span continuous beam according to the present invention.

[0065] Figure 12 This is the bending moment diagram of a unit force acting on a measuring point 3 in a two-span continuous beam according to the present invention.

[0066] Figure 13 This is a model diagram of the five-span continuous beam of the present invention.

[0067] Figure 14 This is a cross-sectional view of the five-span continuous beam T-beam of the present invention. Detailed Implementation

[0068] 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.

[0069] like Figure 1 As shown, a method for identifying single damage in beam structures with different deflections under load is described, with the following specific steps:

[0070] Step 1: Apply load to the beam structure before and after damage, span by span. (a) For simply supported beams, obtain the measured deflection values ​​at two measuring points before and after damage. (b) For continuous beams with three or more spans, obtain the measured deflection values ​​at two measuring points per span and the measured reaction force values ​​at each support before and after damage. (c) For two-span continuous beams, obtain the measured deflection values ​​at two measuring points per span before and after damage. The measured support reaction force values ​​can be measured or not, depending on the actual situation.

[0071] Step 2: Calculate the stiffness EI of the beam structure before damage by using the measured values ​​of the deflection at the measuring points and the measured values ​​of the support reaction force before structural damage.

[0072] 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 in deflection by measuring the deflection values ​​at the measuring points before and after the structural damage, and determine the span in which the damage is located.

[0073] Step 4: Identify the damage location and degree of the beam structure by using the measured values ​​of the deflection at the measuring points after structural damage and the measured values ​​of the reaction force at the supports; Assume that the damaged 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 point is zEI, and the degree of damage is 1-z; (a) For a simply supported beam, since the measured values ​​of the deflection at the two measuring points after structural damage are equal to the calculated values, list the deflection equations of the two measuring points, form a system of equations, and solve for the damage location a and the damage stiffness zEI;

[0074] (b) For continuous beams, select one measuring point 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 support reaction force value to replace the support and obtain the calculated value of the measuring point deflection. Since the measured value of the deflection is equal to the calculated value, set up two sets of equations to solve for the damage location a and the damage stiffness zEI.

[0075] (I) Derivation of the Theory of Simply Supported Beams

[0076] Simply supported beam structure model as follows Figure 2 As shown, the span is L. Assume the damage is located in the left half-span, with a distance of a from the leftmost support. The same applies if the damage is in the right half-span. The damage length is y. The structural stiffness at the undamaged location is EI, and the structural stiffness at the damaged location is zEI (0 < z < 1). Concentrated load... The measurement point is located at the mid-span, measuring point 1 is located at the mid-span, measuring point 2 is located at 0.2L, and measuring point 3 is located at 3 / 4L.

[0077] Apply step 1 to obtain the measured values ​​of deflection at each measuring point before and after damage to the simply supported beam structure.

[0078] (1)

[0079] (2)

[0080] in, , These are the measured values ​​of deflection at point m before and after damage to the i-th span of the beam under a concentrated load applied to the n-span beam. The superscript n indicates the number of spans of the beam, and the subscript... This indicates that a concentrated load is applied to the i-th span, the subscript m indicates the measuring point m, u indicates before structural damage, and d indicates after structural damage.

[0081] Applying step 2, calculate the energy level (EI) of the simply supported beam structure before damage. Concentrated load. The bending moment acting at mid-span is (e.g.) Figure 3 ):

[0082] (3)

[0083] in, The superscript 1 indicates a single-span beam; the first subscript 1 indicates a concentrated load. In the first span, the second 1 indicates the first part.

[0084] The expression for the bending moment when a unit force is applied to measuring point 1 is (e.g.) Figure 4 ):

[0085] (4)

[0086] in, The superscript 1 indicates a single-span beam; the first subscript 1 indicates measuring point 1, and the second subscript 1 indicates the first part.

[0087] The expression for the bending moment when a unit force is applied to measuring point 2 is (e.g.) Figure 5 ):

[0088] (5)

[0089] The expression for deflection is:

[0090] (6)

[0091] Where n represents the number of spans of the beam structure, and i represents the concentrated load. , This represents the actual bending moment of the beam structure. This represents the structural bending moment at measuring point m under a unit force. The actual bending moment of a simply supported beam structure is the same as the bending moment under a concentrated load.

[0092] In summary, the deflection expressions for measuring points 1 and 2 are as follows:

[0093] (7)

[0094] (8)

[0095] EI is obtained from the deflection information before damage, here calculated using the deflection at measuring point 1:

[0096] (9)

[0097] (10)

[0098] By substituting the measured data, the EI can be calculated.

[0099] Applying step 3, for simply supported beams, it is not necessary to determine the number of spans where the damage occurs; there is only one span, and the damage is located in that span.

[0100] Applying step 4, the deflection at two measuring points under concentrated load is 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.

[0101] (11)

[0102] (12)

[0103] The above derivation is based on the assumption that the two measuring points are located on both sides of the damage. Now, we propose an assumption that the two deflection equations are independent and can be solved only when the positions of the two measuring points include the location of the damage. Now, we will conduct a theoretical derivation, in which the two measuring points do not include the damage.

[0104] Applying a unit force at 3 / 4L yields the deflection at that point, which, together with the deflection at mid-span, forms a system of equations. In this case, the location of the damage is not included in the two measurement points.

[0105] (13)

[0106] (14)

[0107] (15)

[0108] As you can see, and The two equations are highly similar, differing only in their proportionality coefficients and constant terms. Therefore, they are related. Solving them under these conditions will yield infinitely many solutions, rather than a stable and unique one. This conclusion can be used to determine the approximate location of the damage. After obtaining the equation system, input data for solving. A stable and unique solution indicates that the damage is included at the locations of the two measuring points; otherwise, it is not. Repeat this process of changing the measuring point locations until a stable and unique solution is obtained, thus completing damage identification. Similarly, applying concentrated loads at two different locations will include the damage location. The deflection at any measuring point under the two load conditions can also be used to solve the problem. If the damage is not included at the locations of the two loads, there will be multiple solutions, and these solutions are often unreasonable.

[0109] 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 6 As 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 .

[0110] 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:

[0111] (16)

[0112] The equivalent stiffness of the two measuring points before damage is:

[0113] (17)

[0114] In the formula:

[0115] (18)

[0116] (19)

[0117] (II) Theoretical Derivation of Two-Span Unequal-Span Continuous Beams

[0118] Two-span unequal span continuous beam structure, such as Figure 7 As shown, a concentrated load is applied at the mid-span of the left span. Apply a concentrated load at the mid-span of the right span. The distance from the leftmost support to the location of the damage is 'a', the damage length is 'y', and the spans of the two spans are 0.4L and 0.6L respectively, with lengths of 'y' and 'y' respectively. and The stiffness of the undamaged part is EI, and the stiffness of the locally damaged element is zEI. Measuring point 1 is located at 0.2L, measuring point 2 is located at 0.4L, measuring point 3 is located at 0.7L, and measuring point 4 is located at 0.9L.

[0119] Apply step 1, as needed, to obtain the measured values ​​of deflection at two measuring points in each span before and after damage to the two-span unequal span continuous beam structure.

[0120] (20)

[0121] (twenty one)

[0122] The basic structural bending moment diagram of the unit force acting on support 2 is as follows: Figure 8 As shown, concentrated load The bending moment diagram of the basic structure of the simply supported beam acting at the mid-span of the left span is as follows: Figure 9 As shown, concentrated load The bending moment diagram of the basic structure of the simply supported beam acting at the mid-span of the right span is as follows: Figure 10 As shown, the bending moment diagram when a unit force is applied to measuring point 1 is as follows. Figure 11As shown, the bending moment diagram when a unit force is applied to measuring point 3 is as follows: Figure 12 As shown.

[0123] The above diagram shows the bending moment of the basic structure under a unit force. The expression is:

[0124] (twenty two)

[0125] A concentrated load is applied at the mid-span of the left span. The corresponding bending moment expression for a simply supported beam is:

[0126] (twenty three)

[0127] A concentrated load is applied at the mid-span of the right span. The corresponding bending moment expression for a simply supported beam is:

[0128] (twenty four)

[0129] A unit force is applied to measuring point 1, which is 0.1L away from support 1. The bending moment expression is:

[0130] (25)

[0131] A unit force is applied to measuring point 3, which is located at the mid-span of the second span. The bending moment expression is:

[0132] (26)

[0133] When the beam is damaged, we can obtain the following using the graphical multiplication method:

[0134] (27)

[0135] (28)

[0136] The concentrated force X can be obtained using the force method and the graphical method. The basic equations can be established using the force method as follows:

[0137] (29)

[0138] (30)

[0139] (31)

[0140] (32)

[0141] 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 system 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.

[0142] Apply step 2 to obtain the EI using the pre-damage information.

[0143] (33)

[0144] (34)

[0145] Substituting the measured data into the equations, the EI before structural damage can be calculated.

[0146] Apply step 3, by comparing the relative difference of the measured deflection values ​​before and after the damage, compare the relative difference of the deflection values ​​before and after the damage at two measuring points in the same span. If they are equal, the damage is not located in this span.

[0147] (35)

[0148] Applying step 4, the deflection expression is listed below:

[0149] (36)

[0150] (37)

[0151] From the above, we can obtain three deflection expressions. Then, by measuring the deflection of three supports in actual engineering, and setting the calculated values ​​equal to the theoretical values, and rearranging EI terms, we get:

[0152] (38)

[0153] (39)

[0154] When using step 3, an index of the relative difference in deflection before and after damage is proposed to distinguish which span the damage is located in, so as to facilitate further analysis, calculate the relative difference in deflection before and after damage at each measuring point, and observe whether they are equal. The proof process is given here.

[0155] (40)

[0156] (41)

[0157] (42)

[0158] (43)

[0159] (44)

[0160] (45)

[0161] (46)

[0162] (47)

[0163] (48)

[0164] (49)

[0165] (50)

[0166] (51)

[0167] (52)

[0168] (53)

[0169] The expression for the relative difference in deflection is:

[0170] (54)

[0171] in, The superscript 2 indicates the number of spans, the subscript i indicates that the load is applied to the i-th span, and m indicates the measuring point m;

[0172] (55)

[0173] (56)

[0174] (57)

[0175] (58)

[0176] (59)

[0177] (60)

[0178] (61)

[0179] (62)

[0180] You can see and Since the values ​​are equal, it can be determined that the damage is not located in the second span, which aligns with the assumption that the damage is located in the first span. Therefore, it is feasible to determine the damaged span by using the relative difference between the measured values ​​of the deflection at the measuring point before and after the damage.

[0181] (III) Theoretical Derivation of n-Span Continuous Beams

[0182] The theoretical derivation of an n-span continuous beam is now performed. Applying step 1, the measured vectors of deflection at the measuring points before and after structural damage and the measured vectors of support reactions are obtained as needed:

[0183] (63)

[0184] (64)

[0185] (65)

[0186] (66)

[0187] Apply step 2 and substitute the measured data to solve for the EI of the beam structure before damage.

[0188] (67)

[0189] (68)

[0190] Apply step 3 to identify which span the damage is located by comparing the relative difference in deflection before and after the damage at the measuring point.

[0191] (69)

[0192] Applying step 4, select one measuring point on each side of the damaged area to measure the deflection, derive its expression, substitute the data, and list the deflection equations to form a system of equations.

[0193] (70)

[0194] (71)

[0195] In this case, assuming that the damage has been determined to be located in the o-th span through step 3, the mid-span measuring points of the o-1 and o+1 spans are selected to formulate the equation. Measuring point (2o-3) is the mid-span measuring point of the (o-1)-th span, and measuring point (2o+1) is the mid-span measuring point of the (o+1)-th span.

[0196] Example 1: Simply supported beam

[0197] Consider a simply supported beam bridge with a span of 100cm. Divide the structure into 20 elements (5cm intervals) 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.

[0198] Use step 1 to obtain information about the beam structure before and after damage.

[0199] Concentrated load The theoretical deflection of measuring point 1 is 0.0035226827, and the simulated value through ANSYS is -0.0035226827. With 10 decimal places 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.

[0200] Concentrated load The theoretical deflection of measuring point 2 is 0.0020251813, and the simulated value through ANSYS is -0.0020251813. With 10 decimal places retained, the simulated value is equal to the actual value.

[0201] Using step 3, substitute the data to obtain the EI.

[0202] (72)

[0203] Step 3 is not required for simply supported beams.

[0204] Using step 4, since the deflection 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 deflection can be listed, forming a nonlinear equation system. (Since the bending moment is chosen to be positive at the top, the calculated value and the measured value are opposites and add up to zero.)

[0205] (73)

[0206] The system of equations is now solved using mathematical software, specifically a nonlinear equation system solver. Since it's a nonlinear system, an iterative method is required, necessitating an initial value. A loop is written to increment the initial value of equation 'a' from 0.05 to 0.5, each increment representing the length of the damage. The initial value of 'z' is arbitrary; we'll assume it's 0.8. First, we solve for the case where 'y' is 5 cm. The results are shown in Table 1.

[0207] Table 1. Measuring points 1 and 2 of the simply supported beam (damaged beam segment length is 0.05m)

[0208]

[0209] As can be seen, by retaining four decimal places, this method can achieve extremely accurate damage localization and quantification, and the solution is the true value. Changing the initial value of z to 0.3 does not change the solution result; therefore, in this method, the given initial value z has little impact on the solution.

[0210] The solution is obtained using measuring points 1 and 3, both of which are located to the right of the damage.

[0211] (74)

[0212] The solution results are shown in Table 2. As can be seen, the damage was not included in the two locations, resulting in multiple solutions and making damage identification impossible.

[0213] Table 2. Measuring points 1 and 3 of the simply supported beam (damaged beam segment length is 0.05m)

[0214]

[0215] 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 3:

[0216] Table 3 Damage identification results of simply supported beams (damage length of beam segment is 0.1m)

[0217]

[0218] 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.

[0219] (75)

[0220] 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 with different deflections under load has high accuracy in damage location and quantification.

[0221] Example 2: Two-span unequal span continuous beam

[0222] 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 units, resulting in 40 units and 41 nodes. The cross-section is 5cm wide and 3cm high. E = Pa. Now assume that the damage location is 30cm away from support 1, the damage degree is 0.3, and the damage length is 5cm.

[0223] Use step 1 to obtain beam structure information.

[0224] Concentrated load The theoretical deflection of measuring point 1 is -0.0005754384, and the simulated value through ANSYS is 0.0005754384. With 10 decimal places retained, the simulated value is equal to the actual value.

[0225] Concentrated load The theoretical deflection of measuring point 3 is 0.0039365003, and the simulated value through ANSYS is -0.0039365003. With 10 decimal places retained, the simulated value is equal to the actual value.

[0226] Using step 2, substitute the data to obtain the EI.

[0227] (76)

[0228] Using step 3, calculate the relative difference in the measured deflection values ​​of each support before and after damage. The results are shown in Table 4.

[0229] Table 4. Relative differences in measured deflection values ​​of two-span continuous beams

[0230]

[0231] Based on the combined expression and example data, it can be seen that when the load is applied to the first span, the relative differences between the measured values ​​of deflection before and after damage at measuring points 3 and 4 in the second span are equal, indicating that the damage is not located in the second span, but rather in the first span. By applying two concentrated loads and measuring the relative differences between the measured values ​​of deflection before and after damage at two measuring points in each span, it is possible to distinguish which span the damage is located in. Applying concentrated loads is relatively easy to implement in engineering.

[0232] Using step 4, solve the deflection equation. Consider optimizing the equation system by multiplying each equation by EI and moving the position of EI into the numerical term to reduce the difference between the two terms and improve accuracy.

[0233] (77)

[0234] Two deflection equations were obtained, with three unknowns. y is a known quantity that we assumed to be y. The solution results are shown in Table 5.

[0235] Table 5 Damage identification results of two-span beams (damage length of beam segment 0.05m)

[0236]

[0237] As can be seen, the solution accuracy is very high. With four decimal places retained, this method can accurately locate and quantify damage, and the solution result is the true value.

[0238] This allows for accurate damage identification by measuring only the deflection of the beam structure and applying two concentrated loads, which is extremely convenient in engineering and has broad application prospects.

[0239] Example 3: Five-span continuous beam

[0240] Analyzing a five-span continuous beam bridge as a representative of an n-span beam, such as... Figure 13 As shown. There are 2 measuring points per span, for a total of 10 measuring points. The five-span continuous beam has a span of 20m per span, a T-beam cross-section, and uses C50 concrete. 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.

[0241] Assume damage occurs in the third span, with damage location 'a' 45m from the leftmost support 1, and the damage level is 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 30m from support 1 and the rear axle is 35m from support 1.

[0242] Apply step 1 to obtain the information needed to solve the problem.

[0243] Apply step 2 and substitute the data to solve for the EI before damage.

[0244] (78)

[0245] Applying step 3, the number of damaged spans was identified by comparing the relative differences in deflection 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 the other spans. The relative differences in the measured deflection before and after the damage are shown in Table 6.

[0246] Table 6. Relative Differences in Measured Deflection Values ​​of a Five-Span Continuous Beam Before and After Damage

[0247]

[0248] 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, 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; in summary, the damage is located in the third span, which is consistent with the actual assumption.

[0249] Applying step 4, list the deflection equation and solve it using the deflections at two measuring points 3 and 7 on the left and right sides of the damaged span. The solution results are shown in Table 7.

[0250] The measured deflection values ​​at measuring points 3 and 7 are 0.0017684205 and -0.0001999100, respectively.

[0251] Table 7 Damage identification results of a five-span beam (damage length of beam segment is 0.05m)

[0252]

[0253] 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.

[0254] 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 with different deflections under load, characterized in that, The steps include: (1) Apply loads to the beam structure before and after damage, (a) for simply supported beams, obtain the measured deflection values ​​at two measuring points before and after damage; (b) for continuous beams with more than three spans, obtain the measured deflection values ​​at two measuring points per span and the measured reaction values ​​at each support before and after damage; (c) for continuous beams with two spans, obtain the measured deflection values ​​at two measuring points per span before and after damage. (2) The stiffness EI of the beam structure before damage is obtained by measuring the deflection at the measuring points and the support reaction before the beam structure is damaged. (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 in deflection by measuring the deflection values ​​at the measuring points before and after the beam structure is damaged, and determine the number of spans where the damage is located. (4) Identify the damage location and degree of damage of the beam structure by measuring the actual deflection value of the measuring point and the actual reaction value of the support after the beam structure is damaged; 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 point is zEI, and the degree of damage is 1-z; (a) For a simply supported beam, if the measured value of the deflection of the two measuring points after the beam structure is damaged is equal to the calculated value, list the deflection equations of the two measuring points, form a set of equations, and solve for the damage location a and the damage stiffness zEI; (b) For continuous beams, select one measuring point 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 value of the measuring point deflection. Set up two sets of equations based on the equality of the measured and calculated deflection values ​​to solve for the damage location a and the damage stiffness zEI. In step (1), the measured vectors of the beam structure deflection and the measured vectors of the support reaction force before and after damage are obtained as needed: ； ； ； ； in, , These are the measured values ​​of deflection at point m before and after damage to the i-th span of the beam under a concentrated load applied to the n-span beam. The superscript n indicates the number of spans of the beam, and the subscript... This indicates that a concentrated load is applied to the i-th span, the subscript m indicates the measuring point m, u indicates before structural damage, and d indicates after structural damage. , These are the measured values ​​of the reaction force at support j before and after the damage to the i-th span structure under the concentrated load of the n-span beam, respectively. The subscript j indicates the support number. In step (2), the stiffness EI of the beam structure before damage is obtained: From the deflection expression: ； in, The calculated value of the deflection at measuring point m before damage occurs when a concentrated load is applied to the i-th span of the beam. This is the total length of the beam structure; The bending moment obtained after applying a unit force at the m-th measuring point of an n-span beam; Let be the bending moment when an n-span beam load is applied to the i-th span. Substitute the measured deflection data to obtain the stiffness EI of the beam structure before damage; ； In step (3), the relative difference between the measured values ​​of deflection before and after the damage at the measuring point is: ； When the relative difference between the measured deflection values ​​before and after damage at the two measuring points in the span is equal, the damage is not located in this span, that is, when and When the values ​​are equal, the damage is not in the i-th span. Considering the load acting on each span, the number of spans where the damage is located is determined comprehensively. In step (4), two equations concerning the deflection at the measuring point 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 concentrated load is applied to the second span. Measuring point 1 is the measuring point on the left side of the first span, and measuring point 3 is the measuring point in the middle of the second span. The two measuring points are located on the left and right sides of the damage, respectively. Let be the bending moment of the simply supported beam basic system 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: ； ； Assuming that the damage has been determined to be located in the o-th span through step (3), the mid-span measuring points of the o-1 and o+1 spans are selected to write the equation. Then, measuring point 2o-3 is the mid-span measuring point of the o-1 span, and measuring point 2o+1 is the mid-span measuring point of the o+1 span. 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, the two unknowns of the two equations can be solved to obtain a and z.

2. The method for identifying single damage in beam structures based on load-induced deflection differences 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 deflection differences 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 process ends when the equation system has a continuous stable solution.

4. The method for identifying single damage in beam structures based on load-induced deflection differences 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 deflection differences according to claim 1, characterized in that: In step (4), if the load is applied once, the equations are solved using the deflections of two measuring points, and the two measuring points must be located on both sides of the damage location; if the load is applied twice, the equations are solved using the deflections of two working conditions of one measuring point, and the two loading positions must be located on both sides of the damage location.

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