CN113310649B - Test method for predicting modal deflection of middle and small bridges - Google Patents
Test method for predicting modal deflection of middle and small bridges Download PDFInfo
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Abstract
The invention discloses a test method for predicting modal deflection of a medium-small bridge. Firstly, arranging a plurality of acceleration sensors on a bridge to be tested, measuring the frequency and the vibration mode of the actually measured bridge under environmental excitation, then parking a load vehicle with a certain weight on a bridge vibration mode measuring point, measuring the bridge frequency and the vibration mode after the load vehicle is added under the environmental excitation, calculating the mass normalization coefficient of the bridge actually measured vibration mode before the mass change through a formula, calculating and identifying the modal compliance matrix of the bridge structure, further predicting the actual modal deflection of the bridge under any static load, wherein the predicted modal deflection is the actual deflection of the bridge under the static load. The method has the advantages of convenient and quick operation, no need of installing a displacement measuring device under the bridge, and capability of saving a great deal of manpower, financial resources and material resources.
Description
Technical Field
The invention relates to a test method for predicting modal deflection of a medium-small bridge, and belongs to the field of bridge health monitoring and safety detection evaluation.
Background
At present, along with the development trend of modern transportation structures with large load, high speed, large flow and the like of transportation industry in China, the load pressure of highway bridges is continuously increased, and the functions and safety of the bridges are seriously influenced due to the influence of factors such as rain and snow erosion, sun drying, freeze thawing and the like. Therefore, how to accurately and rapidly evaluate the structural safety of the bridge in the operation state is an urgent problem to be solved at home and abroad. Bridge deflection measurement is widely applied to bridge engineering, and the deflection of a bridge is an important parameter for judging the technical condition of the bridge. Therefore, the static and dynamic deflection values of the bridge are required to be accurately measured in the aspects of bridge detection, dangerous bridge reconstruction, new bridge acceptance and the like.
Environmental excitation is the most commonly used means for health monitoring at present, and the method relies on natural conditions such as external environment or normal vehicle running to excite, and has the advantage of convenient operation. However, environmental stimuli can only output basic modal parameters of a structure such as: frequency, vibration mode, damping, etc., parameters of deep layers of the structure, such as a modal compliance matrix, etc., cannot be obtained.
The traditional environmental excitation test method lacks of structural input load information, so that the deflection of the bridge structure cannot be directly calculated and predicted by using the modal parameters measured by environmental excitation. The existing bridge load test technology needs to install a complex vertical displacement measuring device under the bridge, and needs to operate with the help of a complex finite element model, so that the operation is complex, and the time and the labor are consumed.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a test method which is convenient and quick to operate and can accurately predict the modal deflection of the middle and small bridges without installing a displacement measuring device under the bridge.
The invention is realized by the following technical scheme: a test method for predicting modal deflection of a medium-small bridge is characterized by comprising the following steps: the method comprises the following steps:
(1) Selecting a span main beam to be tested according to the actual service state and the running condition of the middle-small bridge, and estimating the weight of the bridge test span main beam;
(2) According to a possible theoretical vibration mode form of the main girder of the test span, arranging an acceleration sensor on the main girder of the test span, collecting vertical acceleration time-course response data of a main girder measuring point under environmental excitation, and identifying the actual measurement frequency and vibration mode of the bridge through a mode parameter identification method;
(3) A certain number of load vehicles are parked on vibration mode measuring points of the testing span main beam, acceleration time response data of the measuring points of the rear main beam of the vehicle are acquired under the excitation of the environment, and the actual measurement frequency and the vibration mode of the rear bridge of the vehicle are identified through a mode parameter identification method;
(4) Calculating a mass normalization coefficient of each stage of vibration mode actually measured by the main beam through a mass normalization coefficient calculation formula by utilizing the bridge actually measured frequency and vibration mode before and after the loading vehicle is parked, which are obtained in the step 1 and the step 2; the mass normalization coefficient calculation formula is as follows:
wherein 2ω=ω m +ω m '，Δω＝ω m -ω m '；
Wherein: gamma is the normalized coefficient of vibration mode
ΔM is the change of the front and rear mass matrix of the parked load vehicle
ω m For parking the load-carrying vehicle
ω m ' mth order modal frequency after parking load vehicle
φ' m Column vector for m-th order mode shape after parking load-carrying vehicle
φ m Column vector for m-th order mode shape before parking load-carrying vehicle
φ m T Is phi m Is a transposed vector of (2);
(5) The mass normalization vibration mode of the structure is obtained through calculation according to the following formula, and the mass normalization is carried out on the actual measurement vibration mode:
φ mi ＝γ i φ i ,(i＝1,2,3,4…)
wherein: phi (phi) mi Normalized mode column vector for ith order of mass, gamma i For the i-th order quality normalization coefficient, phi i Normalizing the vibration mode column vector for the i-th order of non-quality;
(6) The measured displacement flexibility matrix [ F ] of the bridge is calculated by using the following formula d ]：
[φ mi ]Normalized mode column vector for the ith order of mass, [ phi ] mi ] T Is [ phi ] mi ]Transposed vector omega i Is the i-th order modal frequency;
(7) Predicting and testing the modal deflection of the span main beam under the action of any static test load by using the obtained bridge actual measurement modal flexibility matrix through the following formula, wherein the predicted modal deflection is the actual deflection of the bridge under the action of the static load:
[d]＝[F d ]{f}
where { f } is the static column vector.
According to the method, the medium-small bridge girder mass matrix M is changed, the change delta M of the mass matrix and the actual measured bridge modal parameters before and after a bridge deck parks a vehicle are utilized to calculate and identify the modal shape mass normalization coefficient and the modal compliance matrix of an actual bridge, and further the deflection deformation of a bridge structure under a vertical static load is predicted.
Further, in step 3, the total weight of the vehicle is 5% -10% of the total weight of the test span main beam.
Further, the mode parameter identification method is EFDD or polymax or SSI or other mode identification methods.
The beneficial effects of the invention are as follows: according to the invention, a plurality of acceleration sensors are firstly arranged on a bridge to be tested, the frequency and the vibration mode of the bridge are measured under environmental excitation, then a load vehicle with a certain weight is parked on a bridge vibration mode measuring point to serve as an additional mass block of the bridge structure, the bridge frequency and the vibration mode after the mass block is added are tested, a mass normalization coefficient of the bridge measured vibration mode before the mass is changed can be obtained through formula calculation, and then the modal compliance matrix of the bridge structure can be calculated, and further the actual modal deflection of the bridge under any static load effect is predicted. The method has the advantages of convenient and quick operation, no need of installing a displacement measuring device under the bridge, and capability of carrying out additional heavy-load vehicles without completely sealing traffic in actual engineering, namely applying mass blocks, realizing the rigidity assessment of the whole test bridge by using a small number of acceleration sensors, and saving a large amount of manpower, financial resources and material resources. Particularly, the actual mass of the bridge structure is difficult to accurately measure, the mass matrix of the bridge structure is changed by adding the mass block, the mass matrix of the bridge is not required to be measured, a finite element model of the bridge is not required to be established, the modal parameters of the bridge can be actually measured by combining environmental excitation only by measuring the weight of the mass block (vehicle) in advance, the actual flexibility matrix of the bridge structure is identified, and the modal deflection of the bridge under any vertical load is further predicted. The method for predicting the modal deflection of the medium-small bridge is convenient and rapid, and has high accuracy.
Drawings
FIG. 1 is a schematic flow chart of the method of the present invention;
FIG. 2a is a schematic diagram (unit: m) of the positions of the masses in the test beam in the first and third mass conditions in the embodiment;
FIG. 2b is a schematic diagram (unit: m) of the position of the proof mass in the test beam in the second proof mass working condition in the embodiment;
FIG. 3 is a schematic diagram of an acceleration sensor arrangement (unit: m) of a test beam in an embodiment;
FIG. 4a is a graph of the measured vibration pattern of the test beam without the mass applied;
FIG. 4b is a graph of the measured vibration pattern of the test beam during one of the mass conditions;
FIG. 4c is a graph of the measured vibration pattern of the test beam during the second working condition of the mass block;
FIG. 4d is a graph of the measured vibration pattern of the test beam under the third condition of the mass block;
FIG. 5a is a graph of a modal compliance matrix (in m/N) calculated for a test beam during a mass condition;
FIG. 5b is a graph of the modal compliance matrix (in m/N) calculated for the test beam during the second mass run;
FIG. 5c is a graph of the modal compliance matrix (in m/N) calculated for the test beam during the third mass run;
fig. 6 is a test beam test section.
Detailed Description
The invention is further illustrated by the following non-limiting examples:
aiming at the defect that the existing bridge load test technology needs to install a complex vertical displacement measuring device under a bridge, the invention provides a test method for predicting the modal deflection of a medium-sized and small bridge based on the combination of environmental excitation and a method for adding mass blocks. The method fully utilizes the measured acceleration time-course data of bridge vibration and the characteristic that the total weight of the main girder at the upper part of the middle-small bridge is relatively light, and has the advantages of no need of a complex finite element model and less interference by human factors. According to the invention, firstly, a testing span main beam is selected according to the actual state and the driving condition of the existing bridge, an acceleration sensor arrangement scheme is formulated, vibration acceleration time response data of the testing span main beam under environmental excitation is collected, and basic modal parameters such as frequency, vibration mode and the like of the bridge are identified through a modal identification method. And then adding a mass block at a proper vibration mode measuring point position of the bridge deck, testing and identifying basic modal parameters of the main girder after adding the mass block, carrying out vibration mode quality normalization on the actual measured vibration mode of the bridge, and then calculating an actual measured modal compliance matrix of the bridge so as to further predict the vertical deflection of the bridge structure under a static load working condition.
The method combines three technologies of environmental excitation bridge modal parameter test, mass block change main beam structure mass matrix and modal displacement flexibility matrix identification, and calculates and predicts the actual deflection of the bridge under any load. The method comprises the following specific steps:
(1) According to the actual service state and running condition of the middle and small bridges, selecting a span main beam to be tested, and widely collecting relevant data, mainly comprising completion drawings, construction logs, maintenance data, maintenance reinforcing data and the like of the bridges, and estimating the weight of the bridge test span main beam.
(2) According to a possible theoretical vibration mode form of the main girder of the test span, arranging an acceleration sensor on the main girder of the test span, collecting vertical acceleration time-course response data of a main girder measuring point under environmental excitation, and identifying the actual measurement frequency and vibration mode of the bridge through a mode parameter identification method; the mode parameter identification method can be a mode parameter identification method such as EFDD, polymax, SSI.
(3) And (3) parking a certain number of load vehicles on the vibration type measuring points of the test span main beam, wherein the total mass of the vehicles is about 5% -10% of the mass of the bridge main beam, and the vehicles are stationary and flameout, so that acceleration time response data of the measuring points of the rear main beam of the vehicle are obtained through actual measurement, and the actual measurement frequency and vibration type of the bridge after the vehicle are identified through a EFDD, polymax, SSI modal parameter identification method.
(4) And (3) calculating a quality normalization coefficient of the bridge vibration mode by using the measured frequencies and the vibration modes of the bridge before and after the load vehicle is parked, which are obtained in the step (1) and the step (2).
The mass matrix of the bridge is set as M, the rigidity matrix is set as K, and the basic equation of free vibration dynamics of the bridge structure is setEquation (K-omega) for mode shape can be derived 2 M) Φ=0, then there is a mode shape and frequency for the mth order
Kφ m ＝ω m 2 Mφ m ,(m＝1，2，3，4...) (1)
Phi in m Normalized mode shape, omega, of the mth order of the non-quality m Is the mth order frequency.
After the load vehicle is parked, the mass matrix of the bridge structure is changed, the relative mass change amount is set as delta M, and the M-th order modal frequency and the modal shape of the structure are correspondingly changed into omega m ' and phi m ' at this time, the formula (1) becomes
Kφ m '＝ω m ' 2 (M+ΔM)φ m ',(m＝1，2，3，4…) (2)
Subtracting (1) from formula (2)
K(φ m '-φ m )+ω m ' 2 ΔMφ m '＝Mω m 2 φ m -Mω m ' 2 φ m ',(m＝1，2，3，4…) (3)
If the relative change of the vibration mode is Δφ, there is
φ m '＝Δφ+φ m ,(m＝1，2，3，4…) (4)
Can be pushed out by substituting the formula (4) into the formula (3)
(ω m 2 -ω m ' 2 )Mφ m ＝ω m ' 2 ΔMφ m '-(K-ω m ' 2 M)Δφ,(m＝1，2，3，4...) (5)
Two sides of equation (5) are multiplied by phi m T Can be obtained by
(ω m 2 -ω m ' 2 )φ m T Mφ m ＝ω m ' 2 φ m T ΔMφ m '-φ m T (K-ω m ' 2 M)Δφ,(m＝1，2，3，4...) (6)
If the mass normalized mode of the mth order mode is psi m T Then
ψ m T Mψ m ＝1,(m＝1，2，3，4...) (7)
For the mode shape, it represents the ratio of the vibration displacement of the structure at each vibration test point, and the inherent property of the structure is reflected, so the ratio of the displacement of each test point in the vibration mode vector is fixed. Therefore, the vibration modes of each order of the structure obtained by actual measurement are only different from the mass normalized vibration mode by one vibration mode mass normalized coefficient gamma, and can be set
ψ m ＝γφ m ,(m＝1，2，3，4...) (8)
Both ends of equation (6) are multiplied by gamma 2 Combined type (7) is obtained
ω m 2 -ω m ' 2 ＝γ 2 ω m ' 2 φ m T ΔMφ m '-γ 2 φ m T (K-ω m ' 2 M)Δφ,(m＝1，2，3，4...) (9)
Let 2ω=ω m +ω m '，Δω＝ω m -ω m ' omega is represented by Deltaomega and omega m And omega m ' available
2ω m ＝2ω+Δω,(m＝1，2，3，4…) (10)
2ω m ' 2 ＝2ω-Δω,(m＝1，2，3，4...) (11)
Substituting the formula (10) and the formula (11) into the formula (9), both sides of the formula are divided by ω m ' 2 Can be obtained by
Since 1/(1- χ) can be found from Taylor expansion 1/(1- χ) =1+χ+o (χ) 2 ) Thus, formula (12) can be written as
Assuming Δφ=0, simplified from equation (13) can be obtained
Under the assumption, the calculation formula of the vibration mode normalization coefficient is as follows
Wherein 2ω=ω m +ω m '，Δω＝ω m -ω m '。
Wherein: gamma is the normalized coefficient of vibration mode
ΔM is the change of the front and rear mass matrix of the parked load vehicle
ω m For parking the load-carrying vehicle
ω m ' mth order modal frequency after parking load vehicle
φ' m Column vector for m-th order mode shape after parking load-carrying vehicle
φ m M-th order mode shape column vector for parking load-carrying vehicle
φ m T Is phi m Is a transposed vector of (2);
(5) After the mass normalization coefficient of each order of vibration mode of the actual measurement girder is calculated by the calculation method in the step 4, the mass normalization vibration mode of the structure is calculated by the formula (16):
φ mi ＝γ i φ i ,(i＝1,2,3,4…) (16)
wherein: phi (phi) mi Normalized mode column vector for ith order of mass, gamma i For the i-th order quality normalization coefficient, phi i And normalizing the vibration mode column vector for the ith order of non-quality.
(6) Calculating a displacement compliance matrix of the bridge, and obtaining an actual measurement displacement compliance matrix [ F ] of the bridge by using a formula (17) d ]
In [ phi ] mi ]Normalized mode column vector for the ith order of mass, [ phi ] mi ] T Is [ phi ] mi ]Is a transposed vector of ω i Is the i-th order modal frequency.
(7) Predicting the actual measurement modal deflection of the bridge under any static load test loading working condition by the bridge actual measurement modal compliance matrix obtained in the step 6, and calculating as formula (18)
[d]＝[F d ]{f} (18)
Where { f } is the static column vector.
The predicted modal deflection is the actual measured deflection of the bridge under the action of any actual static load f.
The invention is specifically described below by taking a static loading test of a simple single beam of an actual bridge as an example.
In this example, the test beam is a 16m standard span simply supported hollow plate beam. The test adopts three working conditions of mass blocks, each working condition is provided with different mass blocks at different measuring points, and the mass block arrangement schematic diagram is shown in figures 2a-2 b. And calculating a vibration mode quality normalization coefficient and a flexibility matrix under each working condition, predicting the modal deflection of the 16-meter main beam under the test load, and comparing with the actually measured deflection.
Mass working condition one: respectively placing 1t weight blocks at the 5L/16 and 11L/16 positions of the test beam;
and a working condition of the mass block II: respectively placing 1t weight blocks at the positions of 3L/16, 11L/16 and 13L/16 of the test beam;
and the working condition of the mass block is three: and 2t weight blocks are respectively placed at the 5L/16 and 11L/16 positions of the test beam.
(1) And collecting basic data of a bridge of a 16m test simply supported beam, and preparing a plurality of mass blocks, wherein the mass blocks are iron blocks with the size of 1.24m multiplied by 0.7m multiplied by 0.146m, and the mass of the iron blocks is approximately equal to 1t weight.
(2) An acceleration sensor is arranged. The test beam is a hollow slab single beam, the sensors are arranged at the top of the test beam at equal intervals, the longitudinal direction of the main beam is divided into eight equal parts, 8 measuring points are all arranged, and 8 acceleration sensors are all arranged on the whole test main beam. The acceleration sensor arrangement is shown in fig. 3.
(3) And adding mass blocks on the surface of the main beam according to the working conditions of the three mass blocks, and measuring and identifying the modal parameters of the test beam before and after adding the mass blocks. Measuring the measured acceleration response data of the test beam without applying the mass block under the environmental excitation by using a EFDD, polymax, SSI modal identification method and identifying the first two-order frequencies of the bridge, wherein the measured acceleration response data are 6.861Hz and 25.879Hz respectively; and measuring the actually measured acceleration response data of the test beam from the working condition I to the working condition III of the mass block under the environmental excitation, and identifying the first two-order frequencies of the bridge, namely 6.485Hz and 24.229Hz, 6.488Hz and 23.795Hz, 6.451Hz and 23.176Hz, wherein the front second-order vibration modes of the test beam before and after the mass block is applied are shown in the attached figures 4a-4 d.
(4) And calculating a vibration mode quality normalization coefficient. And (3) utilizing the frequency and the vibration modes which are recognized before and after the mass block is added in the step (3), and carrying out a formula (15) to calculate vibration mode quality normalization coefficients of different mass block working conditions. The first two-order vibration mode mass normalization coefficients from the first working condition of the mass block to the third working condition of the mass block are respectively 1.29 multiplied by 10 -3 And 1.037×10 -3 、1.311×10 -3 And 9.842 ×10 -4 、1.293×10 -3 And 9.703 ×10 -4 . It can be seen that the vibration mode quality normalization coefficient of each working condition of the mass block is different, and the numerical value of the vibration mode quality normalization coefficient is related to the amplitude of the actually measured vibration mode, the environmental excitation intensity and the like.
(5) And identifying an actual measurement structural mode flexibility matrix of the test beam. And (3) carrying out calculation by using the actual measurement vibration mode and the calculated vibration mode quality normalization coefficient in the step (4) in a formula (16) to obtain a front second-order quality normalization vibration mode of the test beam before the mass block is applied, carrying out calculation by using the quality normalization vibration mode and the actual measurement frequency in a formula (17) to obtain an actual mode flexibility matrix of the test beam, and drawing a test Liang Roudu three-dimensional matrix diagram shown in the attached figures 5a-5c for intuitively displaying the matrix, wherein the flexibility unit is m/N.
(6) And designing a static load test scheme of the 16m single-beam test beam. The test beam test sections are 1/4 cross section, 1/2 cross section and 3/4 cross section, and as particularly shown in fig. 6, the vertical concentrated load is applied to the test beam under the control of the loader. And according to the 0-8t hierarchical loading, 2t stepwise loading, stopping loading after loading to 8t, and dividing into 4 loading levels in total.
(7) Predicting the modal deflection of the girder. And calculating the deflection of the test beam under the load action by using a formula (18) according to the test grading load in the 16m static load test scheme, namely, the predicted modal deflection of the test beam. The predicted modal deflection for each test section is detailed in tables 1-3.
(8) In order to illustrate the accuracy of predicting the modal deflection of the main beam based on environmental excitation and adding the mass blocks, the method provided by the invention is used for identifying the predicted modal deflection and comparing the actual measured deflection value of a static load test:
the relative error of the predicted modal deflection and the experimental static deflection of each test section under the working conditions of the three mass blocks is shown in tables 4-6. As shown in the table, the error values are smaller than 10%, and the requirements of actual bridge engineering test precision can be met. Therefore, the method can accurately predict the actual modal deflection of the middle and small bridges.
TABLE 1 Mass Condition-test Beam predicted modal deflection (Unit: mm)
Table 2 Mass Spectrometry two test beams predicting modal deflection (Unit: mm)
TABLE 3 Mass Spectroscopy three test beams predicting modal deflection (Unit: mm)
TABLE 4 relative error of proof mass working condition-test beam prediction modal deflection and test static deflection
TABLE 5 relative error of modal deflection and test static deflection of two test beams under working condition of mass block
Table 6 relative error of modal deflection and test static deflection of three-test beam under working condition of mass block
Other parts in this embodiment are all of the prior art, and are not described herein.
Claims (3)
1. A test method for predicting modal deflection of a medium-small bridge is characterized by comprising the following steps: the method comprises the following steps:
(1) Selecting a span main beam to be tested according to the actual service state and the running condition of the middle-small bridge, and estimating the weight of the bridge test span main beam;
(2) According to a possible theoretical vibration mode form of the main girder of the test span, arranging an acceleration sensor on the main girder of the test span, collecting vertical acceleration time-course response data of a main girder measuring point under environmental excitation, and identifying the actual measurement frequency and vibration mode of the bridge through a mode parameter identification method;
(3) A certain number of load vehicles are parked on vibration mode measuring points of the testing span main beam, acceleration time response data of the measuring points of the rear main beam of the vehicle are acquired under the excitation of the environment, and the actual measurement frequency and the vibration mode of the rear bridge of the vehicle are identified through a mode parameter identification method;
(4) Calculating a mass normalization coefficient of each stage of vibration mode actually measured by the main beam through a mass normalization coefficient calculation formula by utilizing the bridge actually measured frequency and vibration mode before and after the loading vehicle is parked, which are obtained in the step 1 and the step 2; the mass normalization coefficient calculation formula is as follows:
wherein 2ω=ω m +ω m '，Δω＝ω m -ω m '；
Wherein: gamma is the normalized coefficient of vibration mode
ΔM is the change of the front and rear mass matrix of the parked load vehicle
ω m For parking the load-carrying vehicle
ω m ' mth order modal frequency after parking load vehicle
φ' m Column vector for m-th order mode shape after parking load-carrying vehicle
φ m M-th order modal shape column vector before parking a load vehicle block
φ m T Is phi m Is a transposed vector of (2);
(5) The mass normalization vibration mode of the structure is obtained through calculation according to the following formula, and the mass normalization is carried out on the actual measurement vibration mode:
φ mi ＝γ i φ i ,(i＝1,2,3,4…)
wherein: phi (phi) mi Normalized mode column vector for ith order of mass, gamma i For the i-th order quality normalization coefficient, phi i Normalizing the vibration mode column vector for the i-th order of non-quality;
(6) Benefit (benefit)The measured displacement flexibility matrix [ F ] of the bridge is calculated by the following formula d ]：[φ mi ]Normalized mode column vector for the ith order of mass, [ phi ] mi ] T Is [ phi ] mi ]Transposed vector omega i Is the i-th order modal frequency;
(7) Predicting and testing the modal deflection of the span main beam under the action of any static test load by using the obtained bridge actual measurement modal flexibility matrix through the following formula, wherein the predicted modal deflection is the actual deflection of the bridge under the action of the static load:
[d]＝[F d ]{f}
where { f } is the static column vector.
2. The test method for predicting modal deflection of small and medium bridges according to claim 1, characterized in that: in the step 3, the total weight of the vehicle is 5% -10% of the total weight of the test span main beam.
3. The test method for predicting modal deflection of small and medium bridges according to claim 1, characterized in that: the mode parameter identification method is EFDD or polymax or SSI or other mode identification methods.
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