CN109558680A - Bridge multiple target equivalent static wind load calculation method based on POD technology - Google Patents
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Abstract
The bridge multiple target equivalent static wind load calculation method based on POD technology that the present invention relates to a kind of.Firstly, calculating equivalent static wind load matrix FLRC, intrinsic modal matrix ΦLRC, equivalent static wind load base vector { Φ based on intrinsic modal matrixLRC}N×kWith combination coefficient column vector C；Then, according to the equivalent static wind load basis vector { Φ for calculating acquisitionLRC}N×kWith combination coefficient column vector C, multiple target equivalent static wind load matrix is acquiredFinally, to multiple target equivalent static wind load matrixComputational accuracy and distribution rationality assessed.If the reasonability of calculating error and the equivalent static wind load distribution responded at bridge structure key position meets condition simultaneously, then it is assumed that required equivalent static wind load matrixMeet engineering design requirements.The present invention has comprehensively considered LRC method, the feature of Universal method and PSWL method respectively, and the multiple target for realizing bridge structure equivalent static wind load is equivalent.
Description
Technical Field
The invention belongs to the field of civil engineering, and particularly relates to a bridge multi-target equivalent static wind load calculation method based on a POD technology.
Background
Due to the complex calculation of the buffeting response of the bridge structure, the design specification of the current mainstream wind load adopts equivalent static wind load to replace the buffeting response. Wind engineering researchers have proposed a plurality of equivalent static wind load calculation methods aiming at the characteristics of different types of structures, wherein the original and representative important methods mainly include a gust wind load factor method (GLF), a load-response correlation method (LRC), an inertia force wind load method (IWL), an IWL + LRC combination method, a uniform equivalent static wind load method (Universal) and a basic wind load method (PSWL).
The GLF method is to take the product of the gust load factor G and the average wind load as the equivalent static wind load of the structural wind vibration response. The GLF method has clear concept and simple and convenient operation, but can only ensure that the wind vibration response extreme values at the equivalent positions of the structure are equal, and the responses at other positions have errors of different degrees. The LRC method is an accurate method for calculating the equivalent static wind load of the structural background, but actually, the wind vibration response extreme values at equivalent positions can only be guaranteed to be equal, and the responses at non-equivalent positions have errors of different degrees. The IWL method is a method for calculating the structural resonance equivalent static wind load, is mainly applied to a high-rise building structure with a control function of a first-order vibration mode, and is difficult to consider the influence of the coupling of a high-order vibration mode and the vibration mode. The IWL + LRC combination method integrates the advantages of the IWL method and the advantages of the LRC method, but the method is difficult to consider the influence of high-order mode shape and mode shape coupling, and is actually a single-target equivalent method, and the response at a non-equivalent position has errors of different degrees. The Universal method is a multi-target equivalent static wind load calculation method, but the selection of the equivalent static wind load basic vector is usually completed by the judgment of an engineer. The PSWL method is a method for constructing an equivalent static wind load envelope value, the method needs to use the premise that all response envelope values are not overestimated, although the obtained equivalent static wind load distribution can guarantee the calculation accuracy of the structure, the rationality of the equivalent static wind load distribution is not evaluated, and the local response distortion of the structure can be caused in some cases. At present, an IWL method considering a first-order vibration mode of a structure is adopted in 'building structure load specification' (GB50009-2012) of China, and a static gust load method is adopted in 'highway bridge wind resistance design specification' (JTG/TD 60-01-2004). The static gust load method adopts a force equivalence principle, and compared with the main method based on structural response equivalence, the buffeting response calculation error is larger.
The invention provides a bridge structure multi-target equivalent static wind load vector method based on an intrinsic orthogonal decomposition (POD) technology on the basis of the existing research, which comprehensively considers the respective characteristics of an LRC method, a Universal method and a PSWL method, realizes the multi-target equivalent of the bridge structure equivalent static wind load, and simultaneously considers the distribution rationality.
Disclosure of Invention
The invention aims to provide a POD (platform POD) technology-based bridge multi-target equivalent static wind load calculation method, which comprehensively considers the characteristics of an LRC (line replaceable channel) method, a Universal method and a PSWL (particle swarm optimization) method, realizes multi-target equivalent of bridge structure equivalent static wind loads, and considers the distribution rationality.
In order to achieve the purpose, the technical scheme of the invention is as follows: a bridge multi-target equivalent static wind load calculation method based on POD technology includes the steps of firstly, calculating an equivalent static wind load matrix FLRCEigenmode matrix phiLRCEquivalent static wind load base vector { phi ] based on eigenmode matrixLRC}N×kAnd a combined coefficient column vector C; then, according to the equivalent static wind load basic vector { phi ] obtained by calculationLRC}N×kAnd groupCombining the series vectors C to obtain a multi-target equivalent static wind load matrixFinally, for the multi-target equivalent static wind load matrixThe calculation accuracy and the distribution rationality of (2) are evaluated. If the computational error of the response of the key part of the bridge structure and the rationality of the equivalent static wind load distribution simultaneously meet the conditions, the solved equivalent static wind load matrix is considered to beAnd the engineering design requirements are met. In an embodiment of the present invention, the method specifically includes the following steps:
step S1, calculating an equivalent static wind load matrix FLRCEigenmode matrix phiLRCEquivalent static wind load base vector { phi ] based on eigenmode matrixLRC}N×kAnd a combined coefficient column vector C; wherein,
equivalent static wind load matrix FLRC：
According to the relation between load and response, the extreme value is responded by pulsating windFor an equivalent target, obtaining an equivalent static wind load column vector by an LRC method,
in the formula,is a response at the ith node of the structureCalculating an equivalent static wind load column vector as an equivalent target according to an LRC method;RMS value column vector of fluctuating wind load;for fluctuating wind loads and responsesThe symbol '⊙' represents the operation between matrixes as the multiplication of corresponding elements;
forming an N x m-dimensional equivalent static wind load matrix F by using the obtained equivalent static wind load column vectorsLRC，
Eigenmode matrix phiLRC：
Using intrinsic orthogonal decomposition (POD) technique, called singular value decomposition technique for discrete data structure, equivalent static wind load matrix FLRCThe decomposition is carried out, and the decomposition is carried out,
FLRC＝UΣVT(2)
in the formula, U is an AND matrix FLRCIn an N × N orthogonal matrix with the same number of rows, Σ ═ diag (λ)1…λN) Is a non-negative diagonal matrix of dimension Nxm, and1＞λ2＞…＞λNnot less than 0, V is AND matrix FLRCM × m-dimensional orthogonal matrices having the same number of columns; t represents the conjugate transpose of the matrix, and transpose operation is performed on the real matrix;
let phiLRC＝U，Q＝ΣVTThen F isLRCCan be converted into a standard form of POD decomposition to obtain an eigenmode matrix phiLRC；
FLRC＝ΦLRCQ (3)
In the formula, Q is a corresponding main coordinate matrix;
equivalent static wind load base vector { phi ] based on eigenmode matrixLRC}N×k: in the eigenmode matrix phiLRCThe front k-order column vector is used as a basic vector for constructing the multi-target equivalent static wind load;
combined coefficient sequence vector C:
c is a combination coefficient column vector of the equivalent static wind load base vector, and is a k multiplied by 1 order column vector; defining a function f (C) with C as an unknown number,
wherein | × | non-conducting phosphor2Expressing a two-norm, and min (#) represents the minimum value;
step S2, obtaining an equivalent static wind load basic vector { phi ] according to calculationLRC}N×kThe bridge buffeting response extreme value is taken as an equivalent target to obtain
In the formula,the vector is a buffeting response extreme value column vector (not containing an average wind load response), wherein the sigma r is an RMS value column vector corresponding to the buffeting response r; i is an influence function matrix of the response r;the equivalent static wind load base vector is formed by the first k-order column vectors of the eigenmode matrix; c is a combination coefficient column vector of the equivalent static wind load base vector, and is a k multiplied by 1 order column vector;
then, solving an optimal numerical solution of the combination coefficient column vector C according to a least square method criterion and the formula (4); the multi-target equivalent static wind load of the structure can be obtained after the combination coefficient column vector C is obtained,
in the formula,the multi-target equivalent static wind load column vector is obtained;
step S3, obtaining a multi-target equivalent static wind load matrix according to the obtained multiple target equivalent static wind load matrixCalculating the equivalent static wind load precision, and taking the response of the key part of the bridge as a judgment target; the computational error of the response at the ith critical site of the structure is defined as,
in the formula, epsiloniCalculating an error for the buffeting response at the ith key position of the structure;calculating a buffeting response extreme value at the ith key point of the structure according to a random buffeting theory, and simply referring to the accurate value;the buffeting response extreme value at the ith key position of the structure is calculated according to the formula (6), and is simply called a calculated value; epsiloncriDetermining according to engineering experience for calculating an error control value; and if the calculation errors of the responses at the key parts of the structure all satisfy the formula (7), the calculation accuracy of the equivalent static wind load is considered to satisfy the requirement.
In an embodiment of the present invention, in step S3, on the premise that the calculation accuracy is ensured, the distribution rationality of the equivalent static wind load needs to be checked:
defining the mean value of equivalent static wind loadAnd standard deviation ofAre respectively as
If the equivalent static wind load distribution at two adjacent points of the structure meets the relation (10), the load at the node is considered to have no sudden change, and the rationality of the equivalent static wind load distribution meets the requirement;
α is an empirical coefficient and is determined according to engineering experience;
and if one of the equivalent static wind load and the equivalent static wind load is not satisfied, the equivalent static wind load needs to be calculated again until the two are satisfied.
In one embodiment of the present invention, the epsiloncri10% can be selected, and 10% -20% can be selected from α.
In one embodiment of the present invention,. epsilon.criAnd α should not be too small.
Compared with the prior art, the invention has the following beneficial effects:
(1) the basis vector method integrates the respective characteristics of the LRC method, the Universal method and the PSWL method; the method has the advantages that the LRC method can reflect main information of pulsating wind load distribution and structural response distribution, the defect that the Universal method selects the basic vector of the equivalent static wind load through personal judgment of engineers is overcome, and the hypothesis that the rationality of the equivalent static wind load distribution and the calculated value are not overestimated are not considered in the PSWL method is increased;
(2) the calculation precision and the rationality of the equivalent static wind load distribution are comprehensively considered, and the wind load calculation in the bridge design is guided more reasonably.
Drawings
FIG. 1 is a flow chart of the present invention.
Fig. 2 is an overall layout view (unit: m) of the cable-stayed bridge.
FIG. 3 is a section (unit: cm) of a standard girder.
Fig. 4 is a finite element model of a cable-stayed bridge.
Fig. 5 is a wind load acting on the main beam structure.
FIG. 6 is a graph showing buffeting displacement response RMS of a main beam of the cable-stayed bridge.
FIG. 7 is an equivalent static wind load eigenmode matrix ΦLRCSchematic (first 5 stages).
FIG. 8 is a multi-target equivalent static wind load distribution.
FIG. 9 shows a comparison of buffeting shift response extremes.
FIG. 10 is a multi-target equivalent static wind load distribution change rate.
In the figure: FIG. 6 is a plot of buffeting response RMS value (a) and main beam position (b); in fig. 7, (a), (b) and (c) are schematic diagrams of eigenmode matrixes in three directions of vertical, horizontal and torsional rotation, respectively; in fig. 8, (a) is a vertical equivalent static wind load distribution diagram, (b) is a horizontal equivalent static wind load distribution diagram, and (c) is a torsional equivalent static wind load distribution diagram; in fig. 9, (a), (b), and (c) are buffeting displacement response extrema in three directions of vertical, horizontal, and torsional, respectively; in fig. 10, (a), (b) and (c) are respectively the multi-target equivalent static wind load change rate in the vertical direction, the horizontal direction and the torsion direction.
Detailed Description
The technical scheme of the invention is specifically explained below with reference to the accompanying drawings.
As shown in FIG. 1, the method for calculating the multi-target equivalent static wind load of the bridge comprises 4 parameters which are respectively an equivalent static wind load matrix FLRCEigenmode matrix phiLRCEquivalent static wind load base vector { phi ] based on eigenmode matrixLRC}N×kAnd a combined coefficient vector C. Wherein:
equivalent static wind load matrix FLRC: according to the relation between load and response, the extreme value is responded by pulsating windFor an equivalent target, obtaining an equivalent static wind load column vector by an LRC method,
in the formula,is a response at the ith node of the structureCalculating an equivalent static wind load column vector as an equivalent target according to an LRC method;RMS value column vector of fluctuating wind load;for fluctuating wind loads and responsesThe symbol '⊙' represents the operation between matrixes as the multiplication of corresponding elements;
forming an N x m-dimensional equivalent static wind load matrix F by using the obtained equivalent static wind load column vectorsLRC，
Eigenmode matrix phiLRC: the load matrix F is processed using an intrinsic orthogonal decomposition (POD) technique, referred to as a Singular Value Decomposition (SVD) technique for discrete data structuresLRCThe decomposition is carried out, and the decomposition is carried out,
FLRC＝UΣVT(2)
in the formula, U is an AND matrix FLRCIn an N × N orthogonal matrix with the same number of rows, Σ ═ diag (λ)1…λN) Is a non-negative diagonal matrix of dimension Nxm, and1＞λ2＞…＞λNnot less than 0, V is AND matrix FLRCM × m-dimensional orthogonal matrices having the same number of columns; the symbol "T" represents the conjugate transpose of the matrix, for a real matrix a transpose operation;
let phiLRC＝U，Q＝ΣVTThen F isLRCCan be converted into a standard form of POD decomposition to obtain an eigenmode matrix phiLRC；
FLRC＝ΦLRCQ (3)
In the formula, Q is a corresponding main coordinate matrix;
based on eigenmode momentsEquivalent static wind load base vector { phi ] of arrayLRC}N×k: in the eigenmode matrix phiLRCThe front k-order column vector is used as a basic vector for constructing the multi-target equivalent static wind load.
Combining coefficient vector C: c is a combination coefficient column vector of the equivalent static wind load base vector and is a k multiplied by 1 order column vector. Defining a function f (C) with C as an unknown number,
wherein | × | non-conducting phosphor2Two norms are shown, and min (×) is the minimum.
In the invention, an equivalent static wind load basic vector { phiLRC}N×kThen, the bridge buffeting response extreme value is taken as an equivalent target to obtain
In the formula,the vector is a buffeting response extreme value column vector (not containing an average wind load response), wherein the sigma r is an RMS value column vector corresponding to the buffeting response r; i is an influence function matrix of the response r;the equivalent static wind load base vector is formed by the first k-order column vectors of the eigenmode matrix; c is a combination coefficient column vector of the equivalent static wind load base vector, and is a k multiplied by 1 order column vector;
and then, solving the optimal numerical solution of the base vector combination coefficient C according to the formula (4) according to the least square method criterion. The multi-target equivalent static wind load of the structure can be obtained after the combination coefficient C is obtained,
in the formula,and the multi-target equivalent static wind load column vector is obtained.
In the invention, the equivalent static wind load is obtainedAnd then calculating the equivalent static wind load precision, and taking the response of the key part of the bridge as a judgment target. The computational error of the response at the ith critical site of the structure is defined as,
in the formula, epsiloniCalculating an error for the buffeting response at the ith key position of the structure;calculating a buffeting response extreme value at the ith key point of the structure according to a random buffeting theory, and simply referring to the accurate value;the buffeting response extreme value at the ith key position of the structure is calculated according to the formula (6), and is simply called a calculated value; epsiloncriFor calculating the error control value, the error control value is determined according to engineering experience and can be generally 10%; and if the calculation errors of the responses at the key parts of the structure all satisfy the formula (7), the calculation accuracy of the equivalent static wind load is considered to satisfy the requirement.
In the invention, the distribution rationality of the equivalent static wind load needs to be checked on the premise of ensuring the calculation precision of the buffeting response of the structure. The equivalent static wind load is obtained equivalently according to the overall response of the structure, and is not local response. If distribution of equivalent static wind loadAnd (3) a violent abrupt change occurs, and the phenomenon of local damage or instability of the component is likely to occur in the static load combination calculation. In order to avoid the above false results, the rationality of the equivalent static wind load distribution must be ensured. Defining the mean value of equivalent static wind loadAnd standard deviation ofRespectively, are as follows,
if the equivalent static wind load distribution at two adjacent points of the structure meets the relation (10), the load at the node is considered to have no sudden change, and the rationality of the equivalent static wind load distribution meets the requirement.
Wherein α is an empirical coefficient, and is generally 10% -20% determined by engineering experience.
And if one of the calculation accuracy and the rationality of the equivalent static wind load distribution is not satisfied, the equivalent static wind load needs to be calculated again until the calculation accuracy and the rationality of the equivalent static wind load distribution are satisfied. It is to be noted thatcriAnd α should not be too small.
The present invention is described in detail below with reference to the drawings, but the present invention is not limited thereto.
Referring to fig. 1-10, and tables 1-4, tables 1-4 are as follows:
TABLE 1 buffeting calculation of key parameters
TABLE 2 Multi-target equivalent static wind load calculation error (unit:%)
TABLE 3 Multi-target equivalent static wind load root mean square
TABLE 4 maximum value of multiple-target equivalent static wind load change rate
Example 1:
the main channel bridge of the east-sea bridge is connected with Shanghai and Yangshan deep-water harbors, and is a single-tower single-cable-plane cable-stayed bridge with a span combination of 73+132+420+132+73 as 830m, and the whole arrangement is shown in figure 2. The main beam is a steel-concrete composite box beam, the beam width is 33m, the central beam height is 4m, and the section of the standard main beam is shown in figure 3; the height of the inverted Y-shaped concrete bridge tower is 148 m. The finite element program used in this example was ANSYS, and a finite element model of a cable-stayed bridge is shown in fig. 4.
The main vibration mode and frequency of the front 20 th order of the main beam of the east-sea bridge are obtained through modal analysis and are used as the basis of buffeting calculation, the wind speed acting on the main beam in the actual analysis mainly considers the average wind speed U, the horizontal pulsating wind speed U (t) in the same direction as the average wind and the vertical pulsating wind speed w (t), the wind load acting on the main beam mainly considers the resistance D, the lift L and the lifting moment M, the deformation in the corresponding direction is p, h and α, and the positive direction of each parameter is shown in figure 5.
And (3) performing coupled buffeting frequency domain analysis on the main beam of the cable-stayed bridge by adopting a calculation model considering self-exciting force and buffeting force simultaneously on the basis of a finite element model of the cable-stayed bridge. The self-excitation force adopts a self-excitation force expression based on 18 flutter derivatives proposed by Scanlan; the buffeting force is expressed in terms of buffeting force by Davenport considering the pneumatic admittance, which is a simplified expression of Liepmann of the Sears function. The pneumatic force parameters in the self-excitation force and buffeting force expressions are obtained through a main beam segment model wind tunnel test; derivative of flutterAndthe flutter derivative is obtained through a segment model vibration measurement wind tunnel testAnd P1 *～P6 *The static force three-component force coefficient and the change rate thereof are obtained by a section model force measurement test, and the data at a wind attack angle of 0 degree are shown in a table 1. Horizontal pulsation wind spectrum S in buffeting calculationuu(n) selecting a Kaimal spectrum; vertical pulsating wind spectrum Sww(n) selecting Lumley-Panofsky to correct the wind spectrum; cross-spectra of horizontal and vertical pulsating wind are according to the literature (SimiuE, Scanlan R H. wind Effects on Structures [ M ] M].JohnWiley&Sons, inc.1996); the spatial correlation takes the form suggested by the Highway bridge wind resistance design Specification (JTG/T D60-01-2004). Other major parameters in the buffeting calculation are shown in table 1, using a 0 ° wind attack angle as an example. Considering the front 20 orders of vibration type of the main beam, the buffeting displacement response RMS value of the main beam is obtained according to the CQC combination mode, as shown in FIG. 6. According to fig. 6, the main span midspan, main span quartet, side span-L2 and side span-R2 midspan of the cable-stayed bridge are respectively taken as key points, as shown in fig. 2. Node1, Node3, Node5 refer to side span-L2, main span and side span-R2 mid-span positions, respectively, Node2 and Node4 refer to main span four-branch point positions。
Determining an equivalent static wind load matrix FLRC. Taking the peak factor g as 3.5 according toObtaining an equivalent static wind load matrix F corresponding to the main beam buffeting displacement response extreme valueLRC。
Determining the eigenmode matrix phiLRC. Determining an equivalent static wind load matrix FLRCThen obtaining an equivalent static wind load eigenmode matrix phi through SVDLRCThe first 5 th order eigenmode distribution is shown in fig. 7.
Determining equivalent static wind load basis vector [ phi ] based on eigenmode matrixLRC}N×k. Taking the eigenmode matrix phiLRCThe front k-order column vectors form an equivalent static wind load base vector.
According to the method, the multi-target equivalent static wind load when the base vector k is 1, 3, 6, 10 and 20 can be obtained respectivelyThe distribution of (c) is shown in fig. 8. And (3) respectively acting the multi-target equivalent static wind load in the graph 8 on the main girder of the cable-stayed bridge to obtain a buffeting response extreme value calculation value, as shown in the graph 9. The calculation error of the dither response is small when k is 10 and 20, and for clarity, only the extreme values of the accurate dither response and the calculated values when k is 1, 3 and 6 are shown in fig. 9.
According to the method, the multi-target equivalent static wind load accuracy is calculated, and when k is 1, 3, 6, 10 and 20, the buffeting response calculation error at the key position is shown in the table 2. When k is 6, the response error at all key positions is not more than 7%, wherein the calculation error at the main span key point is not more than 5%, and the requirement of engineering design can be met.
According to the method of the invention, on the premise of meeting the calculation precision, the rationality of the equivalent static wind load distribution needs to be checked. As can be seen from fig. 8, as the number of basis vectors participating in calculation increases, the discreteness of the multi-target equivalent static wind load distribution gradually increases.The root mean square of (d) is shown in table 3. To further evaluate the rationality of the equivalent static wind load distribution, the rate of change of the wind load distribution at different positions was calculated according to equation (10), as shown in fig. 10. The maximum values of the change rates of the equivalent static wind load distributions in each case are shown in table 4. As can be seen from FIG. 10, the change rate of the equivalent static wind load distribution at the side span is obviously greater than that of the main span. When k is not more than 6, the distribution change rate of the equivalent static wind load in the vertical direction, the horizontal direction and the torsion direction is not more than 9%, and the requirement of engineering design can be met.
The above are preferred embodiments of the present invention, and all changes made according to the technical scheme of the present invention that produce functional effects do not exceed the scope of the technical scheme of the present invention belong to the protection scope of the present invention.
Claims (5)
1. A bridge multi-target equivalent static wind load calculation method based on POD technology is characterized by comprising the following steps: firstly, calculating an equivalent static wind load matrix FLRCEigenmode matrix phiLRCEquivalent static wind load base vector { phi ] based on eigenmode matrixLRC}N×kAnd a combined coefficient column vector C; then, according to the equivalent static wind load basic vector { phi ] obtained by calculationLRC}N×kAnd the column vector C of the combination coefficient is summed to obtain a multi-target equivalent static wind load matrixFinally, for the multi-target equivalent static wind load matrixThe calculation accuracy and the distribution rationality of (2) are evaluated. If the computational error of the response of the key part of the bridge structure and the rationality of the equivalent static wind load distribution simultaneously meet the conditions, the solved equivalent static wind load matrix is considered to beAnd the engineering design requirements are met.
2. The method for calculating the bridge multi-target equivalent static wind load based on the POD technology according to claim 1, is characterized in that the method is specifically realized by the following steps:
step S1, calculating an equivalent static wind load matrix FLRCEigenmode matrix phiLRCEquivalent static wind load base vector { phi ] based on eigenmode matrixLRC}N×kAnd a combined coefficient column vector C; wherein,
equivalent static wind load matrix FLRC：
According to the relation between load and response, the extreme value is responded by pulsating windFor an equivalent target, obtaining an equivalent static wind load column vector by an LRC method,
in the formula,is a response at the ith node of the structureCalculating an equivalent static wind load column vector as an equivalent target according to an LRC method;RMS value column vector of fluctuating wind load;for fluctuating wind loads and responsesThe symbol '⊙' represents the operation between matrixes as the multiplication of corresponding elements;
forming an N x m-dimensional equivalent static wind load matrix F by using the obtained equivalent static wind load column vectorsLRC，
Eigenmode matrix phiLRC：
Adopting POD technology, namely singular value decomposition technology for discrete data structure, and equivalent static wind load matrix FLRCThe decomposition is carried out, and the decomposition is carried out,
FLRC＝UΣVT(2)
in the formula, U is an AND matrix FLRCIn an N × N orthogonal matrix with the same number of rows, Σ ═ diag (λ)1… λN) Is a non-negative diagonal matrix of dimension Nxm, and1＞λ2＞…＞λNnot less than 0, V is AND matrix FLRCM × m-dimensional orthogonal matrices having the same number of columns; t represents the conjugate transpose of the matrix, and transpose operation is performed on the real matrix;
let phiLRC＝U，Q＝ΣVTThen F isLRCCan be converted into a standard form of POD decomposition to obtain an eigenmode matrix phiLRC；
FLRC＝ΦLRCQ (3)
In the formula, Q is a corresponding main coordinate matrix;
based on thisEquivalent static wind load base vector { phi ] of sign mode matrixLRC}N×k: in the eigenmode matrix phiLRCThe front k-order column vector is used as a basic vector for constructing the multi-target equivalent static wind load;
combined coefficient sequence vector C:
c is a combination coefficient column vector of the equivalent static wind load base vector, and is a k multiplied by 1 order column vector; defining a function f (C) with C as an unknown number,
wherein | × | non-conducting phosphor2Expressing a two-norm, and min (#) represents the minimum value;
step S2, obtaining an equivalent static wind load basic vector { phi ] according to calculationLRC}N×kThe bridge buffeting response extreme value is taken as an equivalent target to obtain
In the formula,is an extreme column vector of buffeting response, containing no mean wind load response, where σrA column vector of RMS values corresponding to dither response r; i is an influence function matrix of the response r;the equivalent static wind load base vector is formed by the first k-order column vectors of the eigenmode matrix; c is a combination coefficient column vector of the equivalent static wind load base vector, and is a k multiplied by 1 order column vector;
then, solving an optimal numerical solution of the combination coefficient column vector C according to a least square method criterion and the formula (4); the multi-target equivalent static wind load of the structure can be obtained after the combination coefficient column vector C is obtained,
in the formula,the multi-target equivalent static wind load column vector is obtained;
step S3, obtaining a multi-target equivalent static wind load matrix according to the obtained multiple target equivalent static wind load matrixCalculating the equivalent static wind load precision, and taking the response of the key part of the bridge as a judgment target; the computational error of the response at the ith critical site of the structure is defined as,
in the formula, epsiloniCalculating an error for the buffeting response at the ith key position of the structure;calculating a buffeting response extreme value at the ith key point of the structure according to a random buffeting theory, and simply referring to the accurate value;the buffeting response extreme value at the ith key position of the structure is calculated according to the formula (6), and is simply called a calculated value; epsiloncriDetermining according to engineering experience for calculating an error control value; and if the calculation errors of the responses at the key parts of the structure all satisfy the formula (7), the calculation accuracy of the equivalent static wind load is considered to satisfy the requirement.
3. The method for calculating the bridge multi-target equivalent static wind load based on the POD technology according to the claim 2, wherein in the step S3, the distribution rationality of the equivalent static wind load needs to be checked on the premise of ensuring the calculation precision:
defining the mean value of equivalent static wind loadAnd standard deviation ofAre respectively as
If the equivalent static wind load distribution at two adjacent points of the structure meets the relation (10), the load at the node is considered to have no sudden change, and the rationality of the equivalent static wind load distribution meets the requirement;
α is an empirical coefficient and is determined according to engineering experience;
and if one of the equivalent static wind load and the equivalent static wind load is not satisfied, the equivalent static wind load needs to be calculated again until the two are satisfied.
4. The POD technology-based bridge multi-target equivalent static wind load calculation method according to claim 3, wherein epsiloncri10 percent of the total weight of the components and 10 to 20 percent of the total weight of the components α.
5. The POD technology-based bridge multi-target equivalent static wind load calculation method according to claim 3, wherein epsiloncriAnd α should not be too small.
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