CN104457681B - Girder structure dynamic deflection monitoring method based on strain mode - Google Patents

Girder structure dynamic deflection monitoring method based on strain mode Download PDF

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CN104457681B
CN104457681B CN201410478980.6A CN201410478980A CN104457681B CN 104457681 B CN104457681 B CN 104457681B CN 201410478980 A CN201410478980 A CN 201410478980A CN 104457681 B CN104457681 B CN 104457681B
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王佐才
任伟新
耿栋
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Hefei University of Technology
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B21/00—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant
    • G01B21/32—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant for measuring the deformation in a solid

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Abstract

The invention discloses a girder structure dynamic deflection monitoring method based on a strain mode. The method is characterized in that a strain sensor is adopted for carrying out strain testing on a vibrating girder structure to obtain dynamic strain data, a strain mode shape is worked out through a cross-correlation function of the dynamic strain data, a displacement mode shape is worked out through a strain-displacement conversion relation, then, displacement mode coordinates are worked out through the dynamic strain data and the strain mode shape, finally, a real-time deflection line of the girder structure is obtained by superimposing the worked-out displacement mode coordinates on the displacement mode shape, and therefore monitoring of the dynamic deflection is achieved. According to the method, on the premise of guaranteeing precision, the dynamic deflection of the girder structure can be monitored through simple equipment at a low cost.

Description

A kind of girder construction dynamic deflection monitoring method based on strain mode
Technical field
The present invention relates to bridge health monitoring field, specifically a kind of bridge dynamic deflection is monitored based on strain mode Method, monitoring result is used for evaluating use state and the security performance of bridge.
Background technology
It is the important indicator evaluating bridge safty in the deformation of the health monitoring jackshaft girder construction of bridge, be also bridge prison One of key parameter surveyed.Amount of deflection, as a kind of description form of deformation, can set up inner link with the rigidity of bridge, and then Can be with the use state of effective evaluation bridge and security performance.The dynamic deflection of especially bridge, even more bridge force-bearing deform the most Embody in real time.By the monitoring to dynamic deflection, it is possible to obtain the dynamic characteristics of bridge structure and the impact coefficient of vehicular load Deng thus preferably evaluating the health status of bridge, significant to the safe operation ensureing bridge.
Traditional bridge deflection measurement mainly has the methods such as theodolite, spirit level, dial gauge, and these methods extensively should For bridge machinery with monitoring in, but these simple measuring methods are only applicable to the manual measurement of bridge short-term, there is expense When laborious, using inconvenience, the in real time deficiency such as measurement difficulty.In order to realize the real-time monitoring of deflection of bridge span, newer Deflection metrology method such as GPS, laser image, connection tube method, the method for photoelectric imaging etc. gradually emerge, and be applied to bridge In the monitoring of girder construction amount of deflection.But, these measuring methods are required for independent measuring apparatus, increased data volume, increase simultaneously Add the cost of monitoring.For the bridge under particular surroundings, due to complicated operating environment, it is mounted directly displacement The direct measurement that sensor carries out amount of deflection is often very difficult.These methods also can only measure certain or certain finite point simultaneously Deformation, if to obtain whole bridge real-time line of deflection it is necessary to a certain moment measure multiple points deflection value using insert The method of value or matching is obtaining the real-time line of deflection of bridge.But the method for interpolation or matching have ignored bridge structure to exist Vibration effect under Vehicle Load, thus it is difficult to the measurement of the real-time line of deflection of bridge.
Content of the invention
The present invention is for avoiding the weak point existing for above-mentioned prior art, providing a kind of knot of the beam based on strain mode Structure dynamic deflection monitoring method, on the premise of ensureing precision, with relatively low cost and easier mode of operation, realizes to bridge The monitoring of line of deflection in real time.
The present invention is to solve technical problem to adopt the following technical scheme that：
The present invention based on the feature of the girder construction dynamic deflection monitoring method of strain mode is：Using strain transducer to vibration Girder construction carry out strain testing obtain dynamic strain data, strain mode is obtained by the cross-correlation function of described dynamic strain data and shakes Type, obtains the displacement modes vibration shape using strain-displacement transformational relation, is then calculated by dynamic strain data and the strain mode vibration shape again Go out displacement modes coordinate, obtain scratching in real time of girder construction finally according to the displacement modes coordinate superposition displacement modes vibration shape calculating Curve, thus realize the monitoring to dynamic deflection.
The present invention is lain also in based on the feature of the girder construction dynamic deflection monitoring method of strain mode：Described by dynamic strain data Cross-correlation function to obtain the strain mode vibration shape be to carry out as follows：
Assume girder construction dynamic deflection monitoring in, along beam longitudinal direction be evenly arranged M strain transducer correspond to M strain Measuring point, place's girder construction in normal operation is considered as linear structure, according to modal superposition principle, M strain measuring point Strain { ε (t) }M×1As formula (1)：
{ε(t)}M×1=ΨM×N{η(t)}N×1(1)
In formula (1), ΨM×N{ η (t) }N×1It is respectively the strain mode coordinate square of strain mode vibration shape matrix and t Battle array, N is rank number of mode, and has formula (2)：
{η(t)}N×1=GN×M{ε(t)}M×1(2)
In formula (2), GN×MIt is strain weight matrix,Strain weight matrix GN×M's Order is less than use number M of strain transducer, and therefore, N is not more than M；
The zero lag cross-correlation matrix R of strain mode coordinate defines as formula (3)：
R=E [η ηT]=E [G ε εTGT]=GE [ε εT]GT(3)
Element R in cross-correlation matrix R in formula (3)ij(i=1,2 ..., N；J=1,2 ..., N) calculate by formula (4) Arrive：
From the property of strain mode coordinate, if i ≠ j, ηiAnd ηjOrthogonal, Rij=0, cross-correlation matrix R is diagonal Matrix, E [ε εT] it is diagonalization matrix, and G is E [ε εT] eigenvectors matrix；Therefore by matrix E [ε εT] seek feature Vector obtains strain weight matrix GN×M, recycle formula (5) to obtain strain mode vibration shape matrix ΨM×N：
The present invention is also resided in based on the feature of the girder construction dynamic deflection monitoring method of strain mode：Calculate as follows and obtain Obtain the real-time line of deflection of girder construction：
According to modal superposition principle, on beam ordinate, x position is in displacement d (x, t) under t and strain stress (x, t) point Not as formula (6) and formula (7)：
In formula (6) and formula (7), Φi(x) and ΨiX () is respectively the i-th rank displacement modes vibration shape and the i-th rank strain mode is shaken Value at x position for the type, ηiT () is the value in t for the i-th rank modal coordinate；
According to beam principle, the relation of strain-displacement represents as formula (8)：
In formula (8), y (x) is in the distance away from neutral axis for the deck-molding direction for x position, for neutral axis on beam section Position, is tested by parallel strain and obtains；According to formula (8) and formula (7), obtain the i-th rank strain mode vibration shape and the i-th component level shifting formwork Relation between the state vibration shape such as formula (9)：
By being integrated to formula (9), obtain the expression formula such as formula at displacement modes vibration shape x position on beam ordinate (10)：
In formula (10), C1And C2It is integral constant according to determined by the boundary condition of beam, in the sedimentation not considering beam support When, C1And C2Press formula (11) respectively and formula (12) calculates and obtains：
C2=0 (12)
Calculate the matrix form { Φ (x) } obtaining the N rank displacement modes vibration shape by formula (13)1×N：
{Φ(x)}1×N={ Φ1(x),Φ2(x),...Φ,N(x)} (13)
Because displacement modes and strain mode have identical modal coordinate, therefore pass through to calculate the acquisition of strain mode coordinate Displacement modes coordinate, is obtaining strain weight matrix GN×MAfterwards, calculate strain mode coordinate using formula (2)；
According to calculating the displacement modes coordinate obtaining and the displacement modes vibration shape, calculate by formula (6) and obtain the real-time of girder construction Line of deflection d (x, t).
Compared with prior art, the present invention has the beneficial effect that：
1st, the present invention achieves the indirect monitoring to dynamic deflection by testing the dynamic strain of girder construction.In bridge monitoring technology In, bridge strain testing cost is relatively low, is also easier to operate, and also has higher precision.
What the 2nd, the present invention was superimposed that the displacement modes vibration shape obtains according to displacement modes coordinate is that girder construction moves line of deflection, and not only It is the dynamic deflection of certain finite point.
Brief description
Fig. 1 is the floor map of the power hammer hammering lower simply supported beam dynamic deflection monitoring experiment carrying out by the inventive method；
Label in Fig. 1：1 first strain measuring point, 2 second strain measuring points, 3 the 3rd strain measuring points, 4 the 4th strain measuring points, 5 the Five strain measuring points, 6 the 6th strain measuring points, 7 the 7th strain measuring points, 8 the 8th strain measuring points, I first amount of deflection eyeballs, II second scratch Degree eyeball, III the 3rd amount of deflection eyeballs, IV the 4th amount of deflection eyeballs, V the 5th amount of deflection eyeballs；
Fig. 2 is strain-responsive under the random hammering of power hammer for the 4th strain measuring point 4 on beam；
Fig. 3 (a) is the first rank strain mode vibration shape and and theoretical value obtained using the cross-correlation function of dynamic strain data Contrast；
Fig. 3 (b) is the second-order strain mode vibration shape and the contrast with theoretical value obtained；
Fig. 3 (c) is the 3rd rank strain mode vibration shape and the contrast with theoretical value obtained；
Fig. 4 is strain-responsive under the random hammering of power hammer for the 8th strain measuring point 8 on beam；
Fig. 5 (a) is the first rank displacement modes vibration shape obtained using strain-displacement transformational relation and right with theoretical value Than；
Fig. 5 (b) is the second-order displacement modes vibration shape and the contrast with theoretical value obtained；
Fig. 5 (c) is the 3rd rank displacement modes vibration shape and the contrast with theoretical value obtained；
Fig. 6 (a) when power hammering beam 1/4 across when, scratched based on dynamic at the first amount of deflection eyeball I that strain mode calculates Angle value and the contrast with measured value；
Fig. 6 (b) be when power hammer into shape hammering beam 1/4 across when, based at the 3rd amount of deflection eyeball III that strain mode calculates Dynamic deflection value and the contrast with measured value；
Fig. 7 (a) be when power hammer into shape hammering beam 1/8 across when, based on the first amount of deflection eyeball I that strain mode calculates at move Deflection value and the contrast with measured value；
Fig. 7 (b) be when power hammer into shape hammering beam 1/8 across when, based at the 3rd amount of deflection eyeball III that strain mode calculates Dynamic deflection value and the contrast with measured value；
Fig. 8 is the dynamic deflection error being calculated based on strain mode under single-point hammering；
Under Fig. 9 (a) hammers for multipoint random, based on the dynamic deflection at the first amount of deflection eyeball I that strain mode calculates Value and the contrast with measured value；
Fig. 9 (b) is for, under multipoint random hammering, being scratched based on dynamic at the 3rd amount of deflection eyeball III that strain mode calculates Angle value and the contrast with measured value；
Figure 10 is the dynamic deflection error being calculated based on strain mode under multipoint random hammering；
Specific embodiment
In the present embodiment based on the girder construction dynamic deflection monitoring method of strain mode be using strain transducer to vibration Girder construction carries out strain testing and obtains dynamic strain data, obtains strain mode by the cross-correlation function of described dynamic strain data and shakes Type, obtains the displacement modes vibration shape using strain-displacement transformational relation, is then calculated by dynamic strain data and the strain mode vibration shape again Go out displacement modes coordinate, obtain scratching in real time of girder construction finally according to the displacement modes coordinate superposition displacement modes vibration shape calculating Curve, thus realize the monitoring to dynamic deflection.
In being embodied as, obtaining the strain mode vibration shape by the cross-correlation function of dynamic strain data is to carry out as follows：
Girder construction dynamic deflection monitoring in, along beam longitudinal direction be evenly arranged M strain transducer correspond to M strain survey Point, place's girder construction in normal operation is considered as linear structure, and according to modal superposition principle, M strains answering of measuring point Become { ε (t) }M×1As formula (1)：
{ε(t)}M×1=ΨM×N{η(t)}N×1(1)
In formula (1), ΨM×N{ η (t) }N×1It is respectively the strain mode coordinate square of strain mode vibration shape matrix and t Battle array, N is rank number of mode, and has formula (2)：
{η(t)}N×1=GN×M{ε(t)}M×1(2)
In formula (2), GN×MIt is strain weight matrix,Strain weight matrix GN×M's Order is less than use number M of strain transducer, and therefore, N is not more than M.
The zero lag cross-correlation matrix R of strain mode coordinate defines as formula (3)：
R=E [η ηT]=E [G ε εTGT]=GE [ε εT]GT(3)
Element R in cross-correlation matrix R in formula (3)ij(i=1,2 ..., N；J=1,2 ..., N) calculate by formula (4) Arrive：
From the property of strain mode coordinate, if i ≠ j, ηiAnd ηjOrthogonal, Rij=0, cross-correlation matrix R is diagonal Matrix, E [ε εT] it is diagonalization matrix, and G is E [ε εT] eigenvectors matrix；Therefore by matrix E [ε εT] seek feature Vector obtains strain weight matrix GN×M, recycle formula (5) to obtain strain mode vibration shape matrix ΨM×N：
In being embodied as, calculate the real-time line of deflection obtaining girder construction as follows：
According to modal superposition principle, on beam ordinate, x position is in displacement d (x, t) under t and strain stress (x, t) point Not as formula (6) and formula (7)：
In formula (6) and formula (7), Φi(x) and ΨiX () is respectively the i-th rank displacement modes vibration shape and the i-th rank strain mode is shaken Value at x position for the type, ηiT () is the value in t for the i-th rank modal coordinate.
According to beam principle, the relation of strain-displacement represents as formula (8)：
In formula (8), y (x) is in the distance away from neutral axis for the deck-molding direction for x position, for neutral axis on beam section Position, is tested by parallel strain and obtains；According to formula (8) and formula (7), obtain the i-th rank strain mode vibration shape and the i-th component level shifting formwork Relation between the state vibration shape such as formula (9)：
By being integrated to formula (9), obtain the expression formula such as formula at displacement modes vibration shape x position on beam ordinate (10)：
In formula (10), C1And C2It is integral constant according to determined by the boundary condition of beam, in the sedimentation not considering beam support When, C1And C2Press formula (11) respectively and formula (12) calculates and obtains：
C2=0 (12)
Calculate the matrix form { Φ (x) } of the displacement modes vibration shape of the N rank obtaining by formula (13)1×N：
{Φ(x)}1×N={ Φ1(x),Φ2(x),...Φ,N(x)} (13)
Because displacement modes and strain mode have identical modal coordinate, therefore pass through to calculate the acquisition of strain mode coordinate Displacement modes coordinate, is obtaining strain weight matrix GN×MAfterwards, calculate strain mode coordinate using formula (2).
According to calculating the displacement modes coordinate obtaining and the displacement modes vibration shape, calculate by formula (6) and obtain the real-time of girder construction Line of deflection d (x, t).
In this power hammer hammering lower simply supported beam dynamic deflection monitoring experiment, girder construction is freely-supported aluminium beam, long 2.5m, wide 100mm, Thick 20mm.Referring to Fig. 1, it is evenly arranged seven strain gauges in the bottom surface of beam, in corresponding Fig. 1, each strain measuring point is from left to right respectively First strain measuring point 1, second strain measuring point the 2, the 3rd strain measuring point the 3, the 4th strain measuring point the 4, the 5th strain measuring point the 5, the 6th should Become measuring point 6 and the 7th strain measuring point 7, in order to obtain the position of neutral axis on beam section, top surface arranges a strain in girder span Meter, corresponding measuring point is the 8th strain measuring point 8.Because experiment beam is uniform beam, therefore position on beam section for the neutral axis can Obtained by spaning middle section is carried out with parallel strain test.For contrast and experiment, soffit is also evenly arranged five displacements Meter, for testing the actural deflection of beam, is the first amount of deflection eyeball I from left to right respectively, the second amount of deflection eyeball II, and the 3rd scratches Degree eyeball III, the 4th amount of deflection eyeball IV and the 5th amount of deflection eyeball V.
For simply supported beam, the Mode Shape of first three rank is main Mode Shape, and therefore this experiment only considers first three Order mode state.For acquisition strain, the displacement modes vibration shape, firmly hammer carries out hammering at random and gathering each under hammering of multiple spot to beam first The strain-responsive of strain measuring point, the sampling time is 60 seconds, and sample frequency is 200HZ.Fig. 2 show answering of the 4th strain measuring point 4 Become response.The strain mode vibration shape is obtained according to the cross-correlation function of dynamic strain data of the present invention, first three rank obtained should Become Mode Shape and as shown in Fig. 3 (a), Fig. 3 (b) and Fig. 3 (c) respectively with the contrast of theoretical value.For neutral axis on beam section Position, be worth to by the dependent variable calculating the 4th strain measuring point 4 being parallel to each other and the 8th strain measuring point 8, computing formula As follows：
In formula (14), e is the distance away from beam bottom for the neutral axis, ε1And ε2It is respectively the 4th strain measuring point 4 and the 8th strain is surveyed The strain numerical value of point 8, y1And y2It is respectively the 4th strain measuring point 4 and the 8th strain distance away from beam bottom for the measuring point 8.Fig. 4 show Strain-responsive under random hammering for the eight strain measuring points 8.Averaged by multiple calculating, on beam section, neutral axis is away from beam bottom Distance is 10.02mm.After obtaining position of neutral axis, Liang Sicheng Mode Shape is calculated according to formula (10), calculates First three rank displacement modes vibration shape and as shown in Fig. 5 (a), Fig. 5 (b) and Fig. 5 (c) respectively with the contrast of theoretical value.
In order to verify based on strain mode monitor girder construction dynamic deflection method validity, in experiment respectively to beam 1/2 across Place, 1/4 across place, 1/8 hammered across place.Due to only considering first three order mode state, hence with the dynamic strain of three strain measuring points Data and first three rank strain mode vibration shape can calculate first three rank displacement modes coordinate.This experiment utilize second strain measuring point 2, The dynamic strain data that 4th strain measuring point the 4, the 6th strains measuring point 6 calculates displacement modes coordinate, and according to calculated displacement Modal coordinate is superimposed the real-time line of deflection that the displacement modes vibration shape obtains girder construction.The dynamic deflection calculating value is surveyed with displacement meter Dynamic deflection value is contrasted.When power hammer hammering beam 1/4 across when, based at the first amount of deflection eyeball I that strain mode calculates Dynamic deflection value and measured value to such as Fig. 6 (a) Suo Shi, at the 3rd amount of deflection eyeball III, dynamic deflection calculated value and measured value is right Shown in such as Fig. 6 (b).When power hammer hammering beam 1/8 across when, based on dynamic at the first amount of deflection eyeball I that strain mode calculates Deflection value and measured value to such as Fig. 7 (a) Suo Shi, the contrast of dynamic deflection calculated value and measured value at the 3rd amount of deflection eyeball III As shown in Fig. 7 (b).Error is calculated according to the following formula.
In formula (15), E is error criterion, dActual measurementAnd dCalculateIt is respectively measured value and the calculated value of amount of deflection, K is time series Length.In this test, the dynamic deflection error being calculated based on strain mode under single-point hammering within 7%, as Fig. 8 institute Show.
In order to verify the validity calculating girder construction dynamic deflection method based on strain mode further, to click-through multiple on beam Row randomly hammers.Strain the dynamic strain data meter that measuring point the 4, the 6th strains measuring point 6 also with the second strain measuring point the 2, the 4th Calculate displacement modes coordinate, and the displacement modes vibration shape is superimposed according to calculated displacement modes coordinate and obtain scratching in real time of girder construction Curve.The dynamic deflection calculating value is contrasted with displacement meter actual measurement dynamic deflection value.Under multipoint random hammer action, based on should , to such as Fig. 9 (a) Suo Shi, the 3rd amount of deflection is real to become dynamic deflection value and measured value at the first amount of deflection eyeball I that modal calculation goes out At measuring point III, dynamic deflection calculated value and measured value are to such as Fig. 9 (b) Suo Shi.It is based on strain mode under multipoint random hammer action The dynamic deflection error calculating within 6%, as shown in Figure 10.
Experimental verification calculates the validity and accurately of simply supported beam dynamic deflection method under hammer stimulating based on strain mode Property.Based on strain mode, the strain data that the dynamic deflection of girder construction can obtain according only to test calculates acquisition, and has higher Precision.

Claims (2)

1. a kind of girder construction dynamic deflection monitoring method based on strain mode, is characterized in that the beam to vibration using strain transducer Structure carries out strain testing and obtains dynamic strain data, obtains the strain mode vibration shape by the cross-correlation function of described dynamic strain data, Obtain the displacement modes vibration shape using strain displacement transformational relation, then calculated by dynamic strain data and the strain mode vibration shape again Displacement modes coordinate, is superimposed, finally according to the displacement modes coordinate calculating, the real-time flexure that the displacement modes vibration shape obtains girder construction Line, thus realize the monitoring to dynamic deflection；It is by such as that the described cross-correlation function by dynamic strain data obtains the strain mode vibration shape Lower step is carried out：
Assume girder construction dynamic deflection monitoring in, along beam longitudinal direction be evenly arranged M strain transducer correspond to M strain survey Point, place's girder construction in normal operation is considered as linear structure, and according to modal superposition principle, M strains answering of measuring point Become { ε (t) }M×1As formula (1)：
{ε(t)}M×1=ΨM×N{η(t)}N×1(1)
In formula (1), ΨM×N{ η (t) }N×1It is respectively the strain mode coordinates matrix of strain mode vibration shape matrix and t, N is Rank number of mode, and have formula (2)：
{η(t)}N×1=GN×M{ε(t)}M×1(2)
In formula (2), GN×MIt is strain weight matrix,Strain weight matrix GN×MOrder not Exceed use number M of strain transducer, therefore, N is not more than M；
The zero lag cross-correlation matrix R of strain mode coordinate defines as formula (3)：
R=E [η ηT]=E [G ε εTGT]=GE [ε εT]GT(3)
Element R in cross-correlation matrix R in formula (3)ij(i=1,2 ..., N；J=1,2 ..., N) it is calculated by formula (4)：
R i j = E [ η i ( k ) η j ( k ) ] = 1 k lim k → ∞ Σ t = 1 k η i ( t ) η j ( t ) - - - ( 4 )
From the property of strain mode coordinate, if i ≠ j, ηiAnd ηjOrthogonal, Rij=0, cross-correlation matrix R is diagonal matrix, E[εεT] it is diagonalization matrix, and G is E [ε εT] eigenvectors matrix；Therefore by matrix E [ε εT] seek characteristic vector Obtain strain weight matrix GN×M, recycle formula (5) to obtain strain mode vibration shape matrix ΨM×N：
Ψ M × N = ( G N × M T G N × M ) - 1 G N × M T - - - ( 5 ) .
2. the girder construction dynamic deflection monitoring method based on strain mode according to claim 1, is characterized in that：Walk by following The rapid real-time line of deflection calculating acquisition girder construction：
According to modal superposition principle, on beam ordinate, x position is in displacement d (x, t) under t and strain stress (x, t) respectively such as Formula (6) and formula (7)：
d ( x , t ) = Σ i = 1 N Φ i ( x ) η i ( t ) - - - ( 6 )
ϵ ( x , t ) = Σ i = 1 N Ψ i ( x ) η i ( t ) - - - ( 7 )
In formula (6) and formula (7), Φi(x) and ΨiX () is respectively the i-th rank displacement modes vibration shape and the i-th rank strain mode vibration shape in x Value at position, ηiT () is the value in t for the i-th rank modal coordinate；
According to beam principle, the relation of strain-displacement represents as formula (8)：
ϵ ( x , t ) = - y ( x ) ∂ 2 d ( x , t ) ∂ x 2 = - y ( x ) Σ i = 1 N Φ i ′ ′ ( x ) η i ( t ) - - - ( 8 )
In formula (8), y (x) is in the distance away from neutral axis for the deck-molding direction for x position, for position on beam section for the neutral axis, Tested by parallel strain and obtain；According to formula (8) and formula (7), obtain the i-th rank strain mode vibration shape and the i-th rank displacement modes are shaken Relation between type such as formula (9)：
Φ i ′ ′ ( x ) = - 1 y ( x ) Ψ i ( x ) - - - ( 9 )
By being integrated to formula (9), obtain the expression formula such as formula (10) at displacement modes vibration shape x position on beam ordinate：
Φ i ( x ) = ∫ ∫ - 1 y ( x ) Ψ i ( x ) dx 2 + C 1 x + C 2 - - - ( 10 )
In formula (10), C1And C2It is integral constant according to determined by the boundary condition of beam, when not considering the sedimentation of beam support, C1And C2Press formula (11) respectively and formula (12) calculates and obtains：
C 1 = ∫ ∫ 1 y ( x ) Ψ i ( x ) dx 2 | x = l l - - - ( 11 )
C2=0 (12)
Calculate the matrix form { Φ (x) } obtaining the N rank displacement modes vibration shape by formula (13)1×N：
{Φ(x)}1×N={ Φ1(x),Φ2(x),...,ΦN(x)} (13)
Because displacement modes and strain mode have identical modal coordinate, therefore pass through to calculate the acquisition displacement of strain mode coordinate Modal coordinate, is obtaining strain weight matrix GN×MAfterwards, calculate strain mode coordinate using formula (2)；
According to calculating the displacement modes coordinate obtaining and the displacement modes vibration shape, calculate the real-time flexure obtaining girder construction by formula (6) Line d (x, t).
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