CN113761627B - Continuous beam damage identification method based on support counter-force influence line Katz1 fractal dimension - Google Patents
Continuous beam damage identification method based on support counter-force influence line Katz1 fractal dimension Download PDFInfo
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Abstract
The invention discloses a continuous beam damage identification method based on a support counter-force influence line Katz1 fractal dimension, which comprises the following steps: setting support counter-force measuring points at the positions of the supports of the continuous beam, and obtaining support counter-force influence lines of the supports by applying moving loads to the continuous beam before and after damage; solving the Katz1 fractal dimension of the support reaction force influence lines before and after the continuous beam is damaged, and making a difference, and carrying out damage positioning through mutation on the support reaction force influence line Katz1 fractal dimension difference curve; further quantifying the overall damage degree according to the rule of the fractal dimension of the support counter-force influence line Katz1 before and after damage; if the continuous beam is more than two spans, the damage degree is quantified through superposition of support counter-force influence lines Katz1 fractal dimension of a plurality of supports; to obtain a precise degree of damage, the degree of damage of the damaged unit can be further calculated. The invention only needs to set measuring points at each support, saves the consumption of the sensor, can accurately position and quantify the damage of the continuous beam, and is applied to damage assessment of the continuous beam.
Description
Technical Field
The invention relates to the technical field of continuous beam structure damage detection, in particular to a continuous beam damage identification method based on a support counter-force influence line Katz1 fractal dimension.
Background
In recent years, along with the progress of technology and social development of China, traffic becomes more and more convenient, and as bridges serve as throats of traffic, the number of bridges is rapidly increased along with the increase of highway (railway) mileage. The safety problem of the bridge is not only related to traffic, but also closely connected with the economic development of society and the life safety of people. The bridge accident can cause huge property loss and casualties, so that the society can be negatively affected to a certain extent. During the service period of the bridge, the condition that part of bridge structures are damaged due to the influence of artificial or natural factors is unavoidable, so that the state and the performance of the bridge are required to be detected, and the damaged structures or components are reinforced or replaced to ensure that the bridge is in a healthy state. At present, the main methods for identifying structural damage include a dynamic parameter-based method, a static parameter-based damage identification method, a novel intelligent algorithm, model correction and the like, wherein the former two methods are common in the practical application of damage identification. Judging damage of the structure by using factors such as flexibility, damping, vibration mode and the like of the structure based on a dynamic parameter method; static parameter-based methods typically apply a static load to the structure and then identify damage based on factors such as support reaction forces, deflection, and strain. Compared with a method based on dynamic parameters, the method based on static parameters reduces the requirement on the precision of the instrument, is less influenced by external factors, and is mature in technology and equipment; furthermore, methods based on static parameters are also widely studied.
Most methods based on static parameter structural damage identification technology research can only realize the positioning of damage or the qualitative determination of damage degree, but cannot realize the quantification of damage degree. In the structural damage identification research, the Katz fractal dimension is difficult and heavy in solving and deducing damage quantification due to the complexity of a formula, and the continuous beam damage identification method based on the support counter force influence line Katz1 fractal dimension modifies the Katz fractal dimension, so that the quantification of damage can be realized on the premise of having pre-damage information. With the progress of sensor technology, the continuous beam damage identification method based on the support reaction force influence line Katz1 fractal dimension is expected to be applied to damage identification of a structure, and at present, literature reports related to damage identification based on the support reaction force influence line Katz1 fractal dimension are fresh.
Disclosure of Invention
In order to solve the technical problems, the invention provides a continuous beam damage identification method based on a support counter-force influence line Katz1 fractal dimension, which is simple in algorithm and low in cost.
The technical scheme for solving the problems is as follows: the continuous beam damage identification method based on the support counter-force influence line Katz1 fractal dimension is characterized by comprising the following steps of:
(1) Setting support counter-force measuring points at the positions of the supports of the continuous beam, and obtaining support counter-force influence lines of the supports by applying moving loads to the continuous beam before and after damage;
(2) Solving the Katz1 fractal dimension of the support reaction force influence lines before and after the injury, and making a difference, and carrying out injury positioning through mutation on the support reaction force influence line Katz1 fractal dimension difference curve;
the calculation method of the Katz fractal dimension FD comprises the following steps:
wherein, subscripts i, j are node numbers, x i ,y(x i ) Respectively x and y coordinate values of the i node on the curve, d (x i M) is the first point on the curve in the sliding window and othersThe linear distance at the point in the sliding window is the maximum; l (x) i M) represents the total length of the line segment within the sliding window; m represents the scale of the sliding window, i.e. the number of points in the sliding window; l (x) i M) =n×l, where l is the average distance of the distances between adjacent measurement points in the sliding window, and N is the average distance number in the sliding window;
the Katz fractal dimension is centered at the first point in the sliding window, and the modified Katz1 fractal dimensionThe second point in the sliding window is taken as the center, and the calculation method comprises the following steps:
wherein s is an adjusting factor of adjusting coordinate axes y and x units, s is more than 0, and other variables have the same meaning as those in the Katz fractal dimension;
(3) If the structure is a two-span continuous beam structure, the overall damage degree is quantified through the change rule of the fractal dimension of the Katz1 influence line of the counter force of the support before and after damage;
(b) If the continuous beam structure is more than two spans, the integral damage degree quantification is carried out through superposition of fractal dimension of support counter force influence lines Katz1 before and after the damage of a plurality of supports;
(4) In order to further obtain accurate damage degree of the damage unit, the damage degree of the damage part can be quantified according to the Katz1 fractal dimension change rule of the support counter force influence lines before and after the damage of the continuous beam.
In the method for identifying the damage to the continuous beam based on the support reaction force influence line Katz1 fractal dimension, in the step (2), the support reaction force influence line before and after the damage obtains the Katz1 fractal dimension and makes a difference DL as a damage positioning index, and when m=3, the DL is as follows:
when m=4, DL is:
wherein n is the number of nodes, the node 1 is positioned at a support at one end of the beam structure, namely a starting point of application of a moving load, the node n is positioned at a support at the other end of the beam structure, namely a terminal point of application of the moving load, the number of nodes is continuous, the number of nodes is increased from 1 to n in sequence, M is a certain node number, subscripts d and u respectively represent after damage and before damage, when M=3, M is 2, n-1, and when M=4, M is 2, n-2;
when m=3 and m=4, in the calculationThe process comprises the following steps:
wherein,middle y (x) m )＝X dm ，/>Middle y (x) m )＝X um ，x m X is the distance from the starting point of the moving load when the moving load acts on the m node m Is the phase at the supportA corresponding support counter force value;
in the step (2), if the structure is more than two continuous beams, the DL results of a plurality of span supports are combined to judge the damaged position.
In the method for identifying damage to the continuous beam based on the support reaction force influence line Katz1 fractal dimension, in the step (3) (a), when m=3, the method for calculating the quantitative index De of the damage degree of the whole continuous beam is as follows:
De＝[De 2 De 3 … De m … De n-2 De n-1 ]；
in De m The degree of structural damage at the m-node; in order to better explain the damage degree calculation method of each damage position, an i node is provided, and the i node is a node at the left side of the damage unit;
the damage degree calculating method of the structural intermediate unit comprises the following steps:
for the structural side units, i.e. i=1 or i=n-1, the damage degree calculation method is as follows:
(m=2 or m=n-1);
when m=4, the overall damage degree quantitative index De calculation method is as follows:
De＝[De 2 De 3 … De m … De n-3 De n-2 ]；
the damage degree calculation method of the structural intermediate unit is the same as M=3, but the range is M epsilon [2, n-2];
for the structural side units, i.e. i=1 or i=n-1, the damage degree calculation method is as follows:
(m=2 or m=n-2).
In the method for identifying damage to the continuous beam based on the support counter-force influence line Katz1 fractal dimension, in the step (3) (b), quantitative indexes of the damage degree of the whole continuous beam structure with more than two spans are obtainedCompared with De, the calculation method adds a superposition process, and when m=3, no damage of the intermediate support side unit is taken as an example:
wherein,the degree of structural damage at the m-node;
the damage degree calculating method for the structural intermediate unit comprises the following steps:
wherein G represents the number of the supports, G represents the total number of the supports, and the numbers are from 1 to G in sequence;
for the condition of damage of the side unit at the middle support, eliminating the support reaction force influence line data at the support, and superposing and calculating the damage degree by using other support data;
when the structural side unit is damaged, i.e. i=1 or i=n-1, the specific method for calculating the damage degree is as follows:
(m=2 or m=n-1).
In the method for identifying the damage of the continuous beam based on the fractal dimension of the support counter-force influence line Katz1, in the step (4), the overall damage degree quantitative result is not very accurate for the middle unit, and in order to further obtain the accurate damage degree of the damage unit, the damage degree quantitative calculation method of the damage degree of the damage position can be carried out according to the change rule of the fractal dimension of the support counter-force influence line Katz1 before and after the damage of the continuous beam, and only aims at the middle unit when M=3; the accurate result can be obtained through the integral damage degree calculation of the structural edge unit;
the quantitative calculation method of the damage degree of the damaged part comprises the following steps:
wherein, the subscript i represents the node number at the left side of the damaged unit, and i+1 represents the node number at the right side of the damaged unit; wherein, when the damage intermediate unit is not presentOr->When the damage degree is calculated by discarding the value; the influence of factors such as noise, measurement errors and the like is reduced by taking the average value after the damage degree of the support counter force influence lines of the support measuring points is calculated by adopting the method.
In the method for identifying the damage of the continuous beam based on the support reaction force influence line Katz1 fractal dimension, in the step (1), when a moving load is loaded on the continuous beam, the number of nodes of each span is not less than 6 when m=3, the number of nodes of each span is not less than 7 when m=4, and the method comprises two nodes at a support.
The invention has the beneficial effects that: according to the method, a moving load is applied before and after the damage of the continuous beam, so that support counter force influence lines at all supports of the continuous beam are obtained, after the fractal dimension of the support counter force influence lines Katz1 before and after the damage is used for making a difference, damage positioning can be carried out according to the mutation of a curve, and meanwhile, two calculation methods for the damage degree of the structure are provided, wherein one method is a method for identifying the whole damage degree, but the identified damage degree is deviated from the actual damage degree; the other is a damage degree calculating method aiming at a damaged part, the defect of the former method can be well made up, and the calculated damage degree quantitative result is accurate. And by taking multiple damage working conditions into consideration through two-span continuous beams and three-span continuous beam calculation examples, the application value of the continuous beam damage identification method based on the support counter-force influence line Katz1 fractal dimension in continuous beam damage identification is verified, and an effective new method is provided for damage positioning and quantification of the continuous beam.
Drawings
Fig. 1 is a flow chart of the method of the present invention.
Fig. 2 is a schematic diagram of a two-span continuous beam structure according to the present invention.
FIG. 3 is a basic structure of the present inventionBending moment diagram.
FIG. 4 is a basic structure M of the present invention P Bending moment diagram.
FIG. 5 is a schematic diagram of a two-span continuous beam node distribution of the present invention.
Fig. 6 is a finite element model diagram of a two-span continuous beam in accordance with an embodiment of the present invention.
Fig. 7 is a schematic diagram of damage location indicators DL from 1# to 3# under a working condition of m=3 according to an embodiment of the present invention.
Fig. 8 is a schematic diagram of damage location indicators DL from 1# to 3# when two m=3.
Fig. 9 is a schematic diagram of damage location indicators DL from 1# to 3# under a working condition of an embodiment of the present invention, where m=4.
Fig. 10 is a schematic diagram of damage location indicators DL from 1# to 3# when two m=4 in the first working condition according to the embodiment of the present invention.
Fig. 11 is a schematic diagram of the damage degree quantitative index De of 1# to 3# under a working condition of an embodiment of the present invention, where m=3.
Fig. 12 is a schematic diagram of the damage degree quantitative index De of 1# to 3# when two m=3 is used in the first working condition of the embodiment of the present invention.
Fig. 13 is a schematic diagram of the damage degree quantitative index De of 1# to 3# under a working condition of an embodiment of the present invention, where m=4.
Fig. 14 is a schematic diagram of the damage degree quantitative index De of 1# to 3# when two m=4 in the first working condition of the embodiment of the present invention.
Fig. 15 is a finite element model diagram of a two-three span continuous beam in accordance with an embodiment of the present invention.
Fig. 16 is a schematic diagram of a damage location index DL of a first working condition 1# according to an embodiment of the present invention.
Fig. 17 is a schematic diagram of a damage location index DL of a second working condition # 2 according to an embodiment of the present invention.
Fig. 18 is a schematic diagram of a damage location index DL of a first working condition 3# according to an embodiment of the present invention.
Fig. 19 is a schematic diagram of a damage location index DL of a first 4# working condition in the second embodiment of the present invention.
Fig. 20 is a schematic diagram of a damage location index DL of a second condition # 2 in the second embodiment of the present invention.
Fig. 21 is a schematic diagram of a damage location index DL of a second working condition # two 4 according to an embodiment of the present invention.
Fig. 22 is a schematic diagram of a quantitative index De of damage degree of the first working condition 1# according to the second embodiment of the present invention.
Fig. 23 is a schematic diagram of a quantitative index De of damage degree of the second working condition # 2 according to the embodiment of the present invention.
Fig. 24 is a schematic diagram of a quantitative index De of damage degree of the second working condition # 3 according to the embodiment of the present invention.
Fig. 25 is a schematic diagram of a quantitative index De of damage degree of the first 4# working condition according to the embodiment of the present invention.
FIG. 26 is a graph showing quantitative indicators of damage degree after superposition of working conditions 1#, 2# and 4# according to the embodiment of the present inventionIs a schematic diagram of (a).
FIG. 27 is a graph showing quantitative indicators of damage degree after superposition of working conditions II # 1, 2# and 4# according to an embodiment of the present inventionIs a schematic diagram of (a).
Detailed Description
The present invention is further described below with reference to the drawings and examples, wherein like reference numerals in the various drawings refer to the same or similar elements unless otherwise specified.
As shown in fig. 1, the continuous beam damage identification method based on the support reaction force influence line Katz1 fractal dimension specifically comprises the following steps:
1. setting support counter-force measuring points at the positions of the supports of the continuous beam, and obtaining support counter-force influence lines of the supports by applying moving loads to the continuous beam before and after damage;
2. solving the Katz1 fractal dimension of the support reaction force influence lines before and after the injury, and making a difference, and carrying out injury positioning through mutation on the support reaction force influence line Katz1 fractal dimension difference curve;
3. if the structure is a two-span continuous beam structure, the overall damage degree is quantified through the change rule of the fractal dimension of the Katz1 influence line of the counter force of the support before and after damage;
(b) If the continuous beam structure is more than two spans, the integral damage degree quantification is carried out through superposition of fractal dimension of support counter force influence lines Katz1 before and after the damage of a plurality of supports;
4. in order to further obtain accurate damage degree of the damage unit, the damage degree of the damage part can be quantified according to the Katz1 fractal dimension change rule of the support counter force influence lines before and after the damage of the continuous beam.
The application step 1:
(1) Taking the first span damage of the two-span equal-span continuous beam as an example, the structural model is shown in fig. 2, the spans of the two spans are L, A, B and C are three supports of the two-span continuous beam respectively, the distance from a moving load P to a support A is z, the distance from the left side of a damaged position to the support A is a, the length of a damaged area is epsilon, the rigidity of an undamaged area is EI, and the rigidity of the damaged area is kEI; the numbers and letters below the beams in the figure represent node numbers, the starting node number of the moving load P is 1, the ending node number is n, the node numbers are continuous and increase in sequence, and the distances between adjacent nodes are epsilon; wherein the action position of the moving load P is at the m node, the node numbers of the left side and the right side of the damaged area are i and i+1 respectively, and the node number of the middle support B is n 1 ；
Taking the support reaction force influence line of the support B as an example, the support reaction force influence line of the support B can be obtained by a force method and graph multiplication; with a simple beam as a basic structure, when a moving load P acts on the beam, a basic equation can be established by a force method as follows:
δ 11 X+Δ 1P ＝0 (1)
wherein X is the support reaction force delta of the middle support B 11 N when a unit force acts on the support B 1 Node displacement, delta 1P N under the action of load P 1 Displacement of the node.
To calculate delta 11 And delta 1P Drawing a bending moment diagram under the action of unit loadMoment diagram M under the action of moving load P P As particularly shown in fig. 3 and 4. Moment diagram in FIG. 3->The expression of (2) is:
wherein:indicating that the unit load acts on the support B, x E [0, L]When the bending moment is at the x position from the beam support A; />Indicating that a unit load is applied to the support B, x.epsilon.L, 2L]When the bending moment is at the x position from the beam support A;
bending moment diagram M in FIG. 4 P The expression is:
wherein: m is M 1 (x) Indicating that the moving load P acts on the distance support A as z position, x E [0, z]When the distance beam support A is a bending moment at the x position; m is M 2 (x) Indicating that the moving load P acts on the Z position of the distance support A, x is E%z,2L]When the distance beam support a is the bending moment at the x position.
The reaction force influence line of the support B is calculated by graph multiplication:
when a moving load P acts on the left side of the damaged area, delta 1P And the support counter-force influence lines are respectively:
when a moving load P acts on the right side of the damaged area (still within the first span), a 1P And the support counter-force influence lines are respectively:
wherein: x is X (l)dm The support counter force of the support B when the moving load P acts on the m node and is positioned at the left side of the damaged area; x is X (r)dm The support counter force of the support B when the moving load P acts on the m node and is positioned on the right side of the damaged area;
when no damaged area exists in the structure, namely before damage, the k=1 is substituted into the formula (6) or (8) to obtain the support counter force X before damage um The specific expression is:
wherein: x is X um Indicating the application of the moving load P to the m-node before the damage and the support reaction force of the corresponding support B.
And (2) application step:
the fractal dimension is used for describing the irregularity and the disorder degree of the fractal body, various methods for calculating the fractal dimension exist at present, and different fractal dimension calculation methods and different results exist, wherein the Katz fractal dimension calculation method is as follows:
wherein: subscripts i, j are node numbers, x i ,y(x i ) Respectively x and y coordinate values of the i node on the curve, d (x i M) is the maximum value of the linear distance between the first point on the curve in the sliding window and other points in the sliding window; l (x) i M) represents the total length of the line segment within the sliding window; m represents the scale of the sliding window, i.e. the number of points in the sliding window; l (x) i M) =n×l, where l is the average distance of the distances between adjacent measurement points in the sliding window, and N is the average distance number in the sliding window;
because the calculation formula of the Katz fractal dimension is complex, the method for deducing damage quantification on the basis of solving theory is difficult, the Katz fractal dimension takes a first point in a sliding window as a center, the continuous beam damage identification method based on the support reaction influence line Katz1 fractal dimension modifies the Katz fractal dimension, the modified Katz1 fractal dimension takes a second point in the sliding window as a center, and the calculation method comprises the following steps:
wherein s is an adjusting factor for adjusting y and x units, s >0, and the meanings of other variables are the same as those in the Katz fractal dimension;
in equation (14), compared with (y (x) i-1+j )-y(x i-1 )) 2 When the value of s is larger or s 2 (x i-1+j -x i-1 ) 2 When the value is larger, the formula (14) can be simplified according to the rule of infinity equivalent, and the simplifying process of the formula (14) is as follows;
similarly, equation (15) is simplified as follows:
the Katz1 fractal dimension is calculated by the support reaction force influence lines before and after injury, and the poor DL is taken as an injury positioning index, and when M=3, the DL is as follows:
when m=4, DL is:
wherein subscripts d and u represent post-injury and pre-injury, respectively;
when m=3 and m=4, in the calculationThe process comprises the following steps:
wherein,middle y (x) m )＝X dm ，/>Middle y (x) m )＝X um ，x m X is the distance from the starting point of the moving load when the moving load acts on the m node m The corresponding support counter force value at the support.
In order to better understand the distribution rule of the Katz1 fractal dimension on each node, the distribution situation of the nodes is defined as follows, and the distribution situation is specifically shown in fig. 5; wherein the numbers and letters below the beams represent the node numbers, gradually increasing from 1 to n, from left to right. i and i+1 represent nodes on two sides of a damaged cell, f represents a node on the left undamaged cell of the i node, j represents a node on the right undamaged cell of the i+1 node, and n 1 Representing the node at the intermediate support;
according to the principle of infinity equivalent, calculating Katz1 fractal dimension of a middle support counter force influence line before and after continuous beam damage when M=3, and calculating Katz1 fractal dimension of the support counter force influence line before damage by taking a calculation method as an exampleWherein y (x) m )＝X um X is taken as u(m-1) 、X u(m) And X u(m+1) And corresponding x m-1 、x m And x m+1 Substituting into the formula (16) and the formula (17) to obtain +.>
Before injury:
wherein: m is E [2, n 1 -1]，z∈[ε,L-ε]；
The same calculation method as before injury, after injury:
at node f (left side of the lesion cell), f ε [2, i-1], z ε [ ε, a- ε ]:
at inode:
at node i+1:
at node j (right side of the lesion cell), where j ε [ i+2, n 1 -1],z∈[a+2ε,L-ε]：
Substituting the Katz1 fractal dimension for each location into equation (20), it is known from the results that DL is a smooth z-dependent curve at the f and j nodes and at the i and i+1 nodesThe change is made, and the rules of the nodes f and j are not met any more, so that the DL curve can appear in the unsmooth nodes i and i+1, namely two mutation points; therefore, by drawing a DL curve graph, the position of the damage unit can be judged according to the point on the curve where the mutation occurs;
when m=4, the method is similar to the calculation method of the fractal dimension of the support reaction force influence line Katz1 before m=3 injury:
before injury:
wherein: m is E [2, n 1 -2]，z∈[ε,L-2ε]；
After injury:
at node f (left side of the lesion cell), where f ε [2, i-2], z ε [ ε, a-2 ε ]:
at node i-1:
at inode:
at node i+1:
at node j (right side of the lesion cell), where j ε [ i+2, n 1 -2],z∈[a+2ε,L-2ε]：
The damage judgment principle when m=4 and m=3 is the same, and the difference is that when m=4, DL curves have mutation at three node positions i-1, i and i+1, so when m=4, the positions of damage units can be judged according to the points where mutation occurs on the curves.
If the continuous beam structure is more than two spans, the influence of the factors can be avoided by combining the DL results of a plurality of span supports due to the fact that inflection points exist on the support counter-force influence lines and the support counter-force influence lines of the supports far away from the damage are insensitive to the damage and the like, so that the judgment of the damage position is realized. And (3) application step:
the method for quantifying the damage degree can be deduced from the change rule of the Katz1 fractal dimension of the support counter-force influence line before and after damage, and the specific process is as follows:
(a) Two span continuous beam structure, when m=3, the overall damage degree quantitative index De is:
De＝[De 2 De 3 … De m … De n-2 De n-1 ] (33)
wherein: de m The degree of structural damage at the m-node;
(1) When the damage is an intermediate unit, the formula (24) or (25) and the corresponding formula (22) are selected for simplification and derivation, and the formula (25) and the corresponding formula (22) are taken as an example, and epsilon is calculated by a small amount relative to a 3 、a 2 Epsilon and aepsilon 2 Relative to L 3 Also, a small amount is calculated, so that simplification can be achieved, and the simplified result is as follows:
/>
further reduction from formulas (34) and (35) above can yield a quantitative formula for the extent of damage:
de when calculated using equation (36) at other undamaged intermediate cells f ＝De j The overall damage degree can be judged by the method of about 0;
(2) When the damage is a side unit, a=0, the damage unit is beside the support a, and there is no damageSo can only be made from +.1 at node (2 nodes)>To simplify the method for deducing and quantifying the damage degree, the method specifically comprises the following steps:
further reduction from formulas (37) and (38) above can yield a quantitative formula for the extent of damage:
since the set damage is in the first span, and when a=2l-epsilon or a is greater than or equal to L, the damage is in the second span, the specific deduction process is similar to that of the first span; the method for quantifying the edge cell damage when i=n-1 is the same as in equation (39).
When m=4, the overall damage degree quantitative index De is:
De＝[De 2 De 3 … De m … De n-3 De n-2 ] (40)
the damage degree calculation method of the structural intermediate unit is the same as M=3, but the range is (2.ltoreq.m.ltoreq.n-2);
(3) When the damage is an intermediate unit, the formula (30) and the corresponding formula (27) are selected for simplification and derivation, and the method is the same as that when m=3, and the simplified result is as follows:
further reduction from formulas (41) and (42) above can yield a quantitative formula for the extent of damage:
(4) When the lesion is a side unit, a=0 at this time, there is noSo can only be made up of +.>To simplify the method of deriving the quantitative measure of the extent of injury, in which: />
Further reduction from formulas (44) and (45) above can yield a quantitative formula for the extent of damage:
and when a=2l_epsilon or a≡l, the damage is located in the second span, the specific deduction process is similar to the first span; the method for quantifying the edge cell damage at i=n-1 is the same as in equation (46).
The effect of quantifying the damage degree is better when m=3 compared with m=4, and the damage positioning index DL and the overall damage degree quantifying method also show more visual results when m=3, such as two nodes on two sides of a damage unit directly protruding (except for side unit damage) when the damage unit is positioned, and are simpler in theory derivation; in conclusion, the Katz1 fractal dimension at m=3 is more advantageous in lesion recognition.
(b) The continuous beam structure with more than two spans only gives the situation when M=3, and as each support reaction force influence line has an inflection point and the support reaction force influence line of a support far away from the damage can not be sensitive to the damage, the damage degree quantification can be interfered, and the following method is provided for the situation:
in quantitative damage degree, the continuous beam structure with more than two spans has quantitative damage degree indexCompared with De, the calculation method adds a superposition process, and the method can reduce the interference effect caused by the inflection point of the support counter force influence line, and the specific superposition method takes M=3 as an example without the damage of a side unit of the middle support:
wherein,the degree of structural damage at the m-node;
the structural intermediate unit is damaged, and the specific method for calculating the damage degree is as follows:
wherein G represents the number of the supports, G represents the total number of the supports, and the numbers are from 1 to G in sequence;
in the quantitative process of the damage degree, when a unit beside a measuring point at the middle support is damaged, the damage degree quantitative result is interfered by a support reaction force influence line beside the damage, so that the support reaction force influence line data at the support is required to be removed, and the damage degree is calculated by overlapping other data;
the specific method for calculating the damage degree of the structural side unit, namely i=1 or i=n-1, is as follows:
and (4) application step:
in the method for quantifying the damage degree of the whole, when the damage is a side unit, the damage degree obtained by calculation is close to the actual damage degree value because the part which is omitted because of simplification is small, and the error is small; when the damage is an intermediate unit, the amount of the damage is more in the simplification process, so that the calculated damage degree quantitative result is not very accurate, in order to reduce the influence of the factors, a damage degree quantitative formula of the damage can be further deduced according to the fractal dimension change rule of the support counter force influence line Katz1 before and after the damage, and when M=4, the damage degree quantitative method of the damage is only given when M=3 because of the difficult theoretical deduction; the i and i+1 nodes at the damaged cell have the following rules:
the i-1 and i+2 nodes beside the impairment have the following rules:
the method for quantifying the damage degree of the intermediate unit can be deduced by combining the formulas (50) and (51), and the specific damage degree quantifying calculation method is as follows:
wherein when the damaged intermediate unit is located beside the side unit, but is absentOr->When the damage degree is calculated, the non-existing value can be removed, and the damage degree is calculated continuously;
the influence of factors such as noise, measurement error and the like is reduced by taking the average value after the damage degree of the support is calculated by adopting the method.
In step 1, when the moving load is loaded on the continuous beam, the number of nodes of each span is not less than 6 when m=3, and the number of nodes of each span is not less than 7 when m=4, including two nodes at the support.
Embodiment one: referring to fig. 6, two continuous beams are arranged in a span of 50+50cm,5cm divides one unit, 21 nodes, the numbers in the upper row circle are unit numbers, and the lower row numbers are node numbers. The cross-sectional dimensions of the beam were bxh=6cm×3cm, and the modulus of elasticity of the material was 2.7x10 3 MPa, density of 1200kg/m 3 。
Considering damage in actual bridge structures, such as crack generation, material corrosion, etc., generally only causes a reduction in structural rigidity with less impact on the quality of the structure. Therefore, in the finite element modeling process, the damage of the cell is simulated by decreasing the elastic modulus. And building a beam structure model by adopting finite element software. Taking the damage condition of a single unit of the two-span continuous beam as an example, considering that the 6 units in the midspan and the 1 units in the side span are damaged respectively, the damage condition is shown in table 1.
TABLE 1 Single damage Condition of two-span continuous beams
The specific implementation steps are as follows:
step 1: and measuring points are arranged at the positions of the support 1#, the support 2# and the support 3#, and a 1kN moving load is applied to the two spans of continuous beams before and after the damage to obtain a support counter force influence line of the actual measurement of the beam support.
Step 2: solving and differencing the Katz1 fractal dimension of the support reaction force influence lines before and after the injury, and carrying out injury positioning (s=10) through the support reaction force influence line Katz1 fractal dimension difference curve; when m=3, the damage localization map is as shown in fig. 7 and 8; the mutation of the 6 and 7 nodes in fig. 7 under the working condition 1 means that there is damage nearby, and the two points correspond to the theoretical i and i+1 nodes respectively, so that the damage is judged to occur on the 6 unit between the 6 and 7 nodes; working condition 2, in which a mutation occurs in the node 2 can be observed from fig. 8, when the side span unit is damaged, as the DL values of the node 1 and the node n do not exist, only one mutation point exists, and in fig. 8, the node 2 corresponds to the theoretical i+1 node, so that the 1 unit at the node 2 is judged to be damaged;
at m=4, the lesion localization graphs are as in fig. 9 and 10; in the working condition 1, the nodes 5, 6 and 7 are suddenly changed in the figure 9, and the three nodes respectively correspond to the theoretical i-1, i and i+1 nodes, so that the damage of the 6 units is judged; working condition 2, in which the mutation of the node 2 can be observed from the figure 10, when the side span unit is damaged, as DL values of 1, n-1 and n nodes do not exist, the number of the mutation points is only 1, and the node corresponds to the theoretical i+1 node, so that the damage of the node 1 is judged;
and under the two working conditions, the damage position identified by the support counter force influence line Katz1 fractal dimension difference curve is the same as the model damage setting position.
Step 3: the overall damage degree is quantified through the change rule of the fractal dimension of the support counter force influence line Katz1 before and after damage, the working condition 1 is adopted, and the damage degree quantification result is shown in figure 11; working condition 2, the quantitative result of damage degree is shown in figure 12; the quantitative result of the damage degree under the two working conditions is very similar to the set damage degree; however, compared with the side units, the damage quantification effect of the middle unit is poor;
when m=4, the damage degree quantification of the working condition 1 and the working condition 2 is specifically shown in fig. 11 and fig. 12 respectively; from the graph, the damage of the side unit can be observed, and the quantitative result of the damage degree is very similar to the assumed damage degree; while the quantitative effect of m=3 is slightly better than that of m=4 when the intermediate unit is damaged, the damage degree is slightly different from that of the assumed damage degree.
When m=3, the damage degree quantitative method of the damaged portion is further used for solving, and the damage degree quantitative data of the damaged portion under the working condition 1 are given in the following steps:
the quantitative results of the calculated damage degree are shown in the following table 2:
TABLE 2 quantitative results of damage at two-span continuous beam damage
The damage degree calculated by the method is the same as the assumed damage degree according to the D value calculation result, so that the method can make up for the defect of the overall damage degree in quantification and can accurately identify the damage degree of the damaged cell in the middle of the continuous beam.
Embodiment two: referring to fig. 15, the three-span continuous beam span arrangement is 50+75+50cm,5cm dividing one unit, 36 nodes (the numbers in the upper row circle in the figure are the unit numbers, and the lower row numbers are the node numbers). The cross-sectional dimensions of the plate were bxh=6cm×3cm, and the modulus of elasticity of the material was 2.7x10 3 MPa, density of 1200kg/m 3 . The damage condition is shown in table 3 considering that the damage occurs at a plurality of places simultaneously.
TABLE 3 Multi-damage Condition of a three-span continuous Beam
The specific implementation steps are as follows:
step 1: and measuring points are arranged at the positions of the support 1#, the support 2#, the support 3# and the support 4#, and a 1kN moving load is applied to the three-span continuous beam before and after the damage to obtain the support counter force influence line of the actual measurement of the beam support.
Step 2: solving and differencing the Katz1 fractal dimension of the support reaction force influence lines before and after the injury, and carrying out injury positioning (s=10) through the support reaction force influence line Katz1 fractal dimension difference curve; the two working conditions are only analyzed when M=3; the working conditions 1, 16 and 17 can obviously observe that the damage units are 5, 18 and 26 units (the 26 unit is a side unit of the middle support), but the mutation at the damage positions of the 18 unit and the 5 unit in the figures 18 and 19 is not obvious, because the damage is far away from the measuring support, and the mutation generated by partial damage largely masks other damage and other reasons, so that the damage identification result is affected. Analyzing the damage positioning result of the span support, for example, adopting the damage positioning results of 1# and 3# which are shown in fig. 16 and 18, and judging that the damage units are 5, 18 and 26 units; in the working condition 2, the damage locating results of the No. 2 and the No. 4 are adopted, as shown in fig. 20 and 21, the damage of the units 5, 18 and 26 can be judged from the two graphs, and the two graphs make up the defect of the damage locating effect of each damage position; therefore, the method can identify the damage position under the condition of multiple damage of the three-span continuous beam.
Step 3: the overall damage degree is quantified through the change of the damage positioning DL value, in the working condition 1, the damage degree of each support is quantitatively calculated, the specific result is shown in fig. 22-25, and the damage degree of a damage unit can be obtained from the graph, but the damage degree is still slightly deviated from the set damage degree; in addition, due to the inflection point, the disturbance of the abnormal peak exists in the De graph, and especially in fig. 24, the recognition of the overall damage degree is directly affected; so the whole damage degree is calculated by adopting a method of overlapping data of other support reaction force influence lines except 3# as shown in fig. 26; although the identification result of the damage degree after superposition is deviated from the set value, the data is more stable and the interference of the peak value is eliminated; in working condition 2, according to working condition 2And (3) the damage positioning result is directly carried out by selecting No. 1, no. 2 and No. 4Calculation, the result is shown in fig. 27, from which it can be seen that there is a deviation from the set value but approaching; thus, the calculation was further performed by using equation (52), which is a method for quantifying the degree of damage at the damaged portion, and the results are shown in Table 4:
TABLE 4 quantitative results of damage at three-span continuous beam damage
In Table 4, from the results of damage quantification from different positions, the damage quantification effect of damage of other units is better than that of the other units beside the middle support; however, from the aspects of a calculation result and a set damage degree, the calculated D value can accurately quantify the damage degree, so that the damage degree of the index on the multi-damage continuous beam can be accurately identified; the two damage degree quantitative indexes of De and D, the former can generally observe the damage approximate range of each damage unit, the latter can give accurate damage quantitative results only aiming at damage positions, and the two indexes can be used according to specific requirements in actual application.
The foregoing description is only of 2 embodiments of the present invention, and all equivalent changes and modifications made according to the claims of the present invention are intended to fall within the scope of the present invention.
Claims (5)
1. A continuous beam damage identification method based on a support counter-force influence line Katz1 fractal dimension is characterized by comprising the following steps:
(1) Setting support counter-force measuring points at the positions of the supports of the continuous beam, and obtaining support counter-force influence lines of the supports by applying moving loads to the continuous beam before and after damage;
(2) Solving the Katz1 fractal dimension of the support reaction force influence lines before and after the injury, and making a difference, and carrying out injury positioning through mutation on the support reaction force influence line Katz1 fractal dimension difference curve;
the calculation method of the Katz fractal dimension FD comprises the following steps:
；
；
；
wherein: subscripts i, j are node numbers,x, y coordinate values of the inode on the curve,/-respectively>The linear distance between the first point on the curve in the sliding window and other points in the sliding window is the maximum value; />Representing the total length of the line segments within the sliding window; m represents the scale of the sliding window, i.e. the number of points in the sliding window; />Wherein l is the average distance between each adjacent measuring point in the sliding window, and N is the average distance number in the sliding window;
the Katz fractal dimension is centered at the first point in the sliding window, and the modified Katz1 fractal dimensionThe second point in the sliding window is taken as the center, and the calculation method comprises the following steps:
；
；
；
wherein s is an adjusting factor of adjusting coordinate axes y and x units, s is more than 0, and other variables have the same meaning as those in the Katz fractal dimension;
(3) If the structure is a two-span continuous beam structure, the overall damage degree is quantified through the change rule of the fractal dimension of the Katz1 influence line of the counter force of the support before and after damage;
(b) If the continuous beam structure is more than two spans, the integral damage degree quantification is carried out through superposition of fractal dimension of support counter force influence lines Katz1 before and after the damage of a plurality of supports;
(4) In order to further obtain accurate damage degree of the damage unit, the damage degree of the damage part can be quantified according to a Katz1 fractal dimension change rule of support counter force influence lines before and after the damage of the continuous beam;
in the step (4), for the intermediate unit, the overall damage degree quantitative result is not very accurate, and in order to further obtain the accurate damage degree of the damage unit, the damage degree of the damage part can be quantified according to the fractal dimension change rule of the support counter force influence line Katz1 before and after the damage of the continuous beam, and the damage degree quantitative calculation method of the damage degree of the damage part is only aimed at the intermediate unit when M=3; the accurate result can be obtained through the integral damage degree calculation of the structural edge unit;
when the intermediate unit is damaged, the damage degree quantitative calculation method of the damaged part comprises the following steps:
；
wherein, the subscript i represents the node number at the left side of the damaged unit, and i+1 represents the node number at the right side of the damaged unit; wherein, when the damage intermediate unit is not presentOr->When the damage degree is calculated by discarding the value; the support counter force influence lines of the support measuring points are averaged after the damage degree is calculated by adopting the method, so that the influence of factors such as noise, measurement error and the like is reduced.
2. The continuous beam damage identification method based on the support reaction force influence line Katz1 fractal dimension according to claim 1, wherein the method is characterized in that: in the step (2), the Katz1 fractal dimension is calculated by a support reaction force influence line before and after injury, and DL with the difference is taken as an injury positioning index, and when m=3, DL is as follows:
；
when m=4, DL is:
；
；
wherein n is the number of nodes, the node 1 is positioned at a support at one end of the beam structure, namely a starting point of application of a moving load, the node n is positioned at a support at the other end of the beam structure, namely a terminal point of application of the moving load, the number of nodes is continuously increased from 1 to n, M is a certain node number, subscripts d and u respectively represent after injury and before injury, when M=3,when m=4, the number of the active groups,；
when m=3 and m=4, in the calculationThe process comprises the following steps:
；
wherein,middle->，/>Middle->，/>For the distance from the starting point of the mobile load when the mobile load acts on the m node, +.>The counter force value of the support is corresponding to the support;
in the step (2), if the structure is more than two continuous beams, the DL results of a plurality of span supports are combined to judge the damaged position.
3. The continuous beam damage identification method based on the support reaction force influence line Katz1 fractal dimension according to claim 1, wherein the method is characterized in that: in the step (3) (a), for the two-span continuous beam structure, when m=3, the method for calculating the quantitative index De of the damage degree of the whole continuous beam is as follows:
；
in the method, in the process of the invention,the degree of structural damage at the m-node; in order to better explain the damage degree calculation method of each damage position, an i node is provided, and the i node is a node at the left side of the damage unit;
the damage degree calculating method of the structural intermediate unit comprises the following steps:
；
to structural edge units, i.e.Or->The damage degree calculating method comprises the following steps:
；
when m=4, the overall damage degree quantitative index De calculation method is as follows:
；
the damage degree calculation method of the structural intermediate unit is the same as M=3, but the range is that；
To structural edge units, i.e.Or->The damage degree calculating method comprises the following steps:
。
4. the continuous beam damage identification method based on the support reaction force influence line Katz1 fractal dimension according to claim 1, wherein the method is characterized in that: in the step (3) (b), quantitative indexes of the damage degree of the whole continuous beam structure with more than two spans are obtainedCompared with De, the calculation method adds the superposition process, and when m=3, no damage of the intermediate support side unit is as follows:
；
wherein,the degree of structural damage at the m-node;
the damage degree calculating method for the structural intermediate unit comprises the following steps:
；
wherein G represents the number of the supports, G represents the total number of the supports, and the numbers are from 1 to G in sequence;
for the condition of damage of the side unit at the middle support, eliminating the support reaction force influence line data at the support, and superposing and calculating the damage degree by using other support data; damage to structural edge units, i.e.Or->The specific method for calculating the damage degree comprises the following steps:
。
5. the continuous beam damage identification method based on the support reaction force influence line Katz1 fractal dimension according to claim 1, wherein the method is characterized in that: in the step (1), when the moving load is loaded on the continuous beam, the number of nodes of each span is not less than 6 when m=3, and the number of nodes of each span is not less than 7 when m=4, including two nodes at the support.
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