CN109558680B - Bridge multi-target equivalent static wind load calculation method based on POD technology - Google Patents

Bridge multi-target equivalent static wind load calculation method based on POD technology Download PDF

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CN109558680B
CN109558680B CN201811451886.6A CN201811451886A CN109558680B CN 109558680 B CN109558680 B CN 109558680B CN 201811451886 A CN201811451886 A CN 201811451886A CN 109558680 B CN109558680 B CN 109558680B
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董锐
李狄钦
翁祥颖
韦建刚
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Abstract

The invention relates to a bridge multi-target equivalent static wind load calculation method based on a POD technology. Firstly, calculating an equivalent static wind load matrix FLRCIntrinsic mode matrix ΦLRCEquivalent 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 matrix
Figure RE-DDA0001963508130000011
Finally, for the multi-target equivalent static wind load matrix
Figure RE-DDA0001963508130000012
The 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 be
Figure RE-DDA0001963508130000013
And the engineering design requirement is met. The invention comprehensively considers the characteristics of the LRC method, the Universal method and the PSWL method, and realizes the multi-target equivalence of equivalent static wind load of the bridge structure.

Description

基于POD技术的桥梁多目标等效静力风荷载计算方法Calculation method of bridge multi-objective equivalent static wind load based on POD technology

技术领域technical field

本发明属于土木工程领域，具体涉及一种基于POD技术的桥梁多目标等效静力风荷载计算方法。The invention belongs to the field of civil engineering, and in particular relates to a method for calculating the multi-objective equivalent static wind load of bridges based on POD technology.

背景技术Background technique

由于桥梁结构抖振响应计算复杂，目前主流风荷载设计规范均采用等效静力风荷载做等代替换。风工程研究者针对不同类型结构的特点提出了许多等效静力风荷载计算方法，其中具有原创性和代表性的重要方法主要有阵风荷载因子法(GLF)、荷载-响应相关法(LRC)、惯性力风荷载法(IWL)、IWL+LRC组合法、统一等效静力风荷载法(Universal)和基本风荷载法(PSWL)。Due to the complex calculation of the buffeting response of bridge structures, the current mainstream wind load design codes all use equivalent static wind loads as equivalent replacements. Wind engineering researchers have proposed many equivalent static wind load calculation methods according to the characteristics of different types of structures, among which the original and representative important methods mainly include Gust Load Factor (GLF) and Load-Response Correlation (LRC). , Inertial force wind load method (IWL), IWL+LRC combination method, unified equivalent static wind load method (Universal) and basic wind load method (PSWL).

GLF法是将阵风荷载因子G与平均风荷载的乘积作为计算结构风振响应的等效静力风荷载。GLF法概念清晰，操作简便，但仅能够保证结构等效位置处的风振响应极值相等，其他位置处的响应均存在不同程度的误差。LRC法是一种计算结构背景等效静力风荷载的精确方法，但实际上也只能保证等效位置处的风振响应极值相等，非等效位置处的响应均存在不同程度的误差。IWL法是计算结构共振等效静力风荷载的方法，主要应用于一阶振型起控制作用的高层建筑结构，较难考虑高阶振型和振型耦合的影响。IWL+LRC组合法综合了IWL法和LRC法的优点，但该方法同样较难考虑高阶振型和振型耦合的影响，实际上也是一种单目标等效方法，非等效位置处的响应存在不同程度的误差。Universal法是一种多目标等效静力风荷载计算方法，但对于等效静力风荷载基本向量的选择通常需要通过工程师的判断来完成。PSWL法是一种构建等效静力风荷载包络值的方法，这种方法需要以所有的响应包络值不被高估为前提，虽然得到的等效静力风荷载分布能够保证结构的计算精度，但没有对等效静力风荷载分布的合理性进行评估，在有些情况下会引起结构局部响应失真。目前我国《建筑结构荷载规范》(GB50009-2012)采用的是考虑结构一阶振型的IWL法，《公路桥梁抗风设计规范》(JTG/TD60-01-2004)采用的是静阵风荷载法。静阵风荷载法采用的是力等效原则，与基于结构响应等效的上述主要方法相比，抖振响应计算误差较大。The GLF method takes the product of the gust load factor G and the average wind load as the equivalent static wind load for calculating the wind-vibration response of the structure. The GLF method has a clear concept and is easy to operate, but it can only ensure that the extreme values of the wind vibration response at the equivalent position of the structure are equal, and the responses at other positions have different degrees of error. The LRC method is an accurate method for calculating the equivalent static wind load of the structural background, but in fact, it can only ensure that the extreme values of the wind vibration response at the equivalent position are equal, and the responses at the non-equivalent positions have different degrees of error. . The IWL method is a method for calculating the equivalent static wind load of structural resonance. It is mainly used in high-rise buildings where the first-order mode shape plays a controlling role, and it is difficult to consider the influence of higher-order mode shapes and mode coupling. The IWL+LRC combination method combines the advantages of the IWL method and the LRC method, but this method is also difficult to consider the influence of high-order mode shapes and mode shape coupling. In fact, it is also a single-target equivalent method. different degrees of error. The Universal method is a multi-objective equivalent static wind load calculation method, but the selection of the basic vector of the equivalent static wind load usually needs to be completed by the engineer's judgment. The PSWL method is a method for constructing equivalent static wind load envelope values. This method requires that all response envelope values are not overestimated. Although the obtained equivalent static wind load distribution can ensure the structural stability The accuracy of the calculations was not evaluated, but the reasonableness of the distribution of equivalent static wind loads was not assessed, which in some cases would distort the local response of the structure. At present, my country's "Code for Loading of Building Structures" (GB50009-2012) adopts the IWL method considering the first-order vibration mode of the structure, and "Code for Design of Highway Bridges against Wind" (JTG/TD60-01-2004) adopts the static gust load method. . The static gust load method adopts the principle of force equivalence. Compared with the above-mentioned main methods based on the equivalence of structural response, the calculation error of buffeting response is larger.

本发明在已有研究基础上提出了一种基于本征正交分解(POD)技术的桥梁结构多目标等效静力风荷载基向量法，该方法综合考虑了LRC法、Universal法和PSWL法各自的特点，实现了桥梁结构等效静力风荷载的多目标等效，同时考虑了其分布的合理性。Based on the existing research, the present invention proposes a multi-objective equivalent static wind load basis vector method for bridge structures based on intrinsic orthogonal decomposition (POD) technology, which comprehensively considers the LRC method, the Universal method and the PSWL method. According to their respective characteristics, the multi-objective equivalence of the equivalent static wind load of the bridge structure is realized, and the rationality of its distribution is also considered.

发明内容SUMMARY OF THE INVENTION

本发明的目的在于提供一种基于POD技术的桥梁多目标等效静力风荷载计算方法，该方法综合考虑了LRC法、Universal法和PSWL法各自的特点，实现了桥梁结构等效静力风荷载的多目标等效，同时考虑了其分布的合理性。The purpose of the present invention is to provide a multi-objective equivalent static wind load calculation method for bridges based on POD technology, which comprehensively considers the respective characteristics of LRC method, Universal method and PSWL method, and realizes the equivalent static wind load of bridge structure The multi-objective equivalence of the load takes into account the rationality of its distribution.

为实现上述目的，本发明的技术方案是：一种基于POD技术的桥梁多目标等效静力风荷载计算方法，首先，计算等效静力风荷载矩阵FLRC、本征模态矩阵ΦLRC、基于本征模态矩阵的等效静力风荷载基向量{ΦLRC}N×k和组合系数列向量C；然后，依据计算获得的等效静力风荷载基本向量{ΦLRC}N×k和组合系数列向量C，求得多目标等效静力风荷载矩阵

Figure BDA0001886622100000021
最后，对多目标等效静力风荷载矩阵
Figure BDA0001886622100000022
的计算精度和分布合理性进行评估。若桥梁结构关键部位处响应的计算误差和等效静力风荷载分布的合理性同时满足条件，则认为所求的等效静力风荷载矩阵
Figure BDA0001886622100000023
满足工程设计要求。在本发明一实施例中，该方法具体实现步骤如下：In order to achieve the above purpose, the technical solution of the present invention is: a method for calculating the multi-objective equivalent static wind load of bridges based on POD technology. First, the equivalent static wind load matrix F LRC and the eigenmode matrix Φ LRC are calculated. , based on the eigenmode matrix equivalent static wind load basis vector {Φ LRC } N×k and the combination coefficient column vector C; then, based on the calculated equivalent static wind load basis vector {Φ LRC } N× k and the column vector C of the combined coefficients to obtain the multi-objective equivalent static wind load matrix
Figure BDA0001886622100000021
Finally, for the multi-objective equivalent static wind load matrix
Figure BDA0001886622100000022
The computational accuracy and distribution rationality are evaluated. If the calculation error of the response at the key parts of the bridge structure and the rationality of the equivalent static wind load distribution meet the conditions at the same time, it is considered that the required equivalent static wind load matrix
Figure BDA0001886622100000023
Meet engineering design requirements. In an embodiment of the present invention, the specific implementation steps of the method are as follows:

步骤S1、计算等效静力风荷载矩阵FLRC、本征模态矩阵ΦLRC、基于本征模态矩阵的等效静力风荷载基向量{ΦLRC}N×k和组合系数列向量C；其中，Step S1, calculate the equivalent static wind load matrix F LRC , the eigenmode matrix Φ LRC , the equivalent static wind load basis vector {Φ LRC } N×k and the combination coefficient column vector C based on the eigenmode matrix ;in,

等效静力风荷载矩阵FLRC：Equivalent static wind load matrix F LRC :

依据荷载-响应之间的关系，以脉动风响应极值

Figure BDA0001886622100000024
为等效目标，通过LRC法获得等效静力风荷载列向量，According to the load-response relationship, the extreme value of the pulsating wind response
Figure BDA0001886622100000024
For the equivalent target, the column vector of the equivalent static wind load is obtained by the LRC method,

Figure BDA0001886622100000025
Figure BDA0001886622100000025

式中，

Figure BDA0001886622100000026
是以结构第i节点处的响应
Figure BDA0001886622100000027
为等效目标，按照LRC法计算出的等效静力风荷载列向量；
Figure BDA0001886622100000028
为脉动风荷载的RMS值列向量；
Figure BDA0001886622100000029
为脉动风荷载与响应
Figure BDA00018866221000000210
之间的荷载-响应相关系数列向量；符号“⊙”表示矩阵之间的运算为对应元素相乘；In the formula,
Figure BDA0001886622100000026
is the response at the ith node of the structure
Figure BDA0001886622100000027
is the equivalent target, the column vector of the equivalent static wind load calculated according to the LRC method;
Figure BDA0001886622100000028
is the column vector of the RMS value of the fluctuating wind load;
Figure BDA0001886622100000029
for fluctuating wind loads and responses
Figure BDA00018866221000000210
The column vector of the load-response correlation coefficient between ; the symbol "⊙" indicates that the operation between the matrices is the multiplication of the corresponding elements;

将获得的等效静力风荷载列向量组成N×m维的等效静力风荷载矩阵FLRC，

Figure BDA00018866221000000211
The obtained equivalent static wind load column vectors are formed into an N×m-dimensional equivalent static wind load matrix F LRC ,
Figure BDA00018866221000000211

本征模态矩阵ΦLRC：The eigenmode matrix Φ LRC :

采用本征正交分解(POD)技术，对于离散数据结构称为奇异值分解技术，对等效静力风荷载矩阵FLRC进行分解，Using the intrinsic orthogonal decomposition (POD) technology, which is called singular value decomposition technology for discrete data structures, the equivalent static wind load matrix F LRC is decomposed,

FLRC＝UΣVT (2)F LRC = UΣV T (2)

式中，U为与矩阵FLRC的行数相同的N×N维正交矩阵，Σ＝diag(λ1…λN)为N×m维非负对角矩阵，且λ1＞λ2＞…＞λN≥0，V为与矩阵FLRC的列数相同的m×m维正交矩阵；T表示矩阵的共轭转置，对于实矩阵为转置运算；In the formula, U is an N×N-dimensional orthogonal matrix with the same number of rows as the matrix F LRC , Σ=diag(λ 1 …λ N ) is an N×m-dimensional non-negative diagonal matrix, and λ 1 >λ 2 >...>λ N ≥ 0, V is an m×m-dimensional orthogonal matrix with the same number of columns as the matrix F LRC ; T represents the conjugate transpose of the matrix, which is a transpose operation for real matrices;

令ΦLRC＝U，Q＝ΣVT，则FLRC可转化为POD分解的标准形式，从而得到本征模态矩阵ΦLRC；Let Φ LRC =U, Q=ΣV T , then F LRC can be transformed into the standard form of POD decomposition, thereby obtaining the eigenmode matrix Φ LRC ;

FLRC＝ΦLRCQ (3)F LRC = Φ LRC Q (3)

式中，Q为对应的主坐标矩阵；In the formula, Q is the corresponding principal coordinate matrix;

基于本征模态矩阵的等效静力风荷载基向量{ΦLRC}N×k：以本征模态矩阵ΦLRC的前k阶列向量作为构建多目标等效静力风荷载的基本向量；Equivalent static wind load basis vector based on eigenmode matrix {Φ LRC } N×k : The first k-order column vector of eigenmode matrix Φ LRC is used as the basic vector for constructing multi-objective equivalent static wind load ;

组合系数列向量C：Combined coefficient column vector C:

C为等效静力风荷载基向量的组合系数列向量，为k×1阶列向量；定义以C为未知数的函数f(C)，C is the column vector of the combination coefficient of the base vector of the equivalent static wind load, which is a column vector of order k × 1; the function f(C) with C as the unknown is defined,

Figure BDA0001886622100000031
Figure BDA0001886622100000031

式中，||*||2表示二范数，min(*)表示求最小值；In the formula, ||*|| 2 represents the second norm, and min(*) represents the minimum value;

步骤S2、依据计算获得的等效静力风荷载基本向量{ΦLRC}N×k，以桥梁抖振响应极值为等效目标，得Step S2, according to the basic vector of equivalent static wind load {Φ LRC } N×k obtained by calculation, and taking the extreme value of the buffeting response of the bridge as the equivalent target, obtain:

Figure BDA0001886622100000032
Figure BDA0001886622100000032

式中，

Figure BDA0001886622100000033
为抖振响应极值列向量(不含平均风荷载响应)，其中σr为抖振响应r对应的RMS值列向量；I为响应r的影响函数矩阵；
Figure BDA0001886622100000034
为本征模态矩阵的前k阶列向量组成的等效静力风荷载基向量；C为等效静力风荷载基向量的组合系数列向量，为k×1阶列向量；In the formula,
Figure BDA0001886622100000033
is the column vector of the extreme value of the buffeting response (excluding the average wind load response), where σr is the column vector of the RMS value corresponding to the buffeting response r; I is the influence function matrix of the response r;
Figure BDA0001886622100000034
is the equivalent static wind load base vector composed of the first k-order column vectors of the eigenmode matrix; C is the combination coefficient column vector of the equivalent static wind load base vector, which is a k×1-order column vector;

而后，将组合系数列向量C根据最小二乘法准则按照式(4)求得最优数值解；结构的多目标等效静力风荷载可在求得组合系数列向量C后求得，Then, the optimal numerical solution is obtained by using the combination coefficient column vector C according to formula (4) according to the least squares criterion; the multi-objective equivalent static wind load of the structure can be obtained after the combination coefficient column vector C is obtained,

Figure BDA0001886622100000035
Figure BDA0001886622100000035

式中，

Figure BDA0001886622100000036
为多目标等效静力风荷载列向量；In the formula,
Figure BDA0001886622100000036
is the column vector of the multi-objective equivalent static wind load;

步骤S3、依据求得的多目标等效静力风荷载矩阵

Figure BDA0001886622100000037
计算等效静力风荷载精度，以桥梁关键部位处的响应作为评判目标；定义结构第i个关键部位处响应的计算误差为，Step S3, based on the obtained multi-objective equivalent static wind load matrix
Figure BDA0001886622100000037
To calculate the equivalent static wind load accuracy, the response at the key part of the bridge is used as the evaluation target; the calculation error of the response at the i-th key part of the structure is defined as,

Figure BDA0001886622100000041
Figure BDA0001886622100000041

式中，εi为结构第i个关键位置处的抖振响应计算误差；

Figure BDA0001886622100000042
为按照随机抖振理论计算得到的结构第i个关键点处的抖振响应极值，简称为精确值；
Figure BDA0001886622100000043
为根据式(6)计算得到的结构第i个关键位置处的抖振响应极值，简称为计算值；εcri为计算误差控制值，根据工程经验确定；若结构关键部位处响应的计算误差均满足式(7)，则认为等效静力风荷载的计算精度满足要求。where ε i is the calculation error of the buffeting response at the ith key position of the structure;
Figure BDA0001886622100000042
is the extreme value of the chattering response at the ith key point of the structure calculated according to the random chattering theory, referred to as the exact value for short;
Figure BDA0001886622100000043
is the extreme value of the buffeting response at the ith key position of the structure calculated according to formula (6), referred to as the calculated value for short; ε cri is the control value of the calculation error, which is determined according to engineering experience; if the calculation error of the response at the key position of the structure is If all satisfy formula (7), it is considered that the calculation accuracy of the equivalent static wind load meets the requirements.

在本发明一实施例中，步骤S3中，等效静力风荷载在保证计算精度的前提下，须验算其分布的合理性：In an embodiment of the present invention, in step S3, under the premise of ensuring the calculation accuracy of the equivalent static wind load, the rationality of its distribution must be checked:

定义等效静力风荷载的均值

Figure BDA0001886622100000044
和标准差
Figure BDA0001886622100000045
分别为Defining the mean value of the equivalent static wind load
Figure BDA0001886622100000044
and standard deviation
Figure BDA0001886622100000045
respectively

Figure BDA0001886622100000046
Figure BDA0001886622100000046

Figure BDA0001886622100000047
Figure BDA0001886622100000047

若结构相邻两点处的等效静力风荷载分布满足关系式(10)，则认为该节点处的荷载不存在突变，等效静力风荷载分布的合理性满足要求；If the distribution of the equivalent static wind load at two adjacent points of the structure satisfies the relation (10), it is considered that there is no sudden change in the load at the node, and the rationality of the distribution of the equivalent static wind load meets the requirements;

Figure BDA0001886622100000048
Figure BDA0001886622100000048

式中，α为经验系数，根据工程经验确定；In the formula, α is the empirical coefficient, which is determined according to engineering experience;

等效静力风荷载的计算精度和等效静力风荷载分布的合理性须同时满足，有其一不满足时，则需返回重新计算等效静力风荷载，直至两者满足。The calculation accuracy of the equivalent static wind load and the rationality of the equivalent static wind load distribution must be satisfied at the same time.

在本发明一实施例中，所述εcri可以取10％，α可以取10％-20％。In an embodiment of the present invention, the ε cri can take 10%, and the α can take 10%-20%.

在本发明一实施例中，εcri和α的取值不应同时过小。In an embodiment of the present invention, the values of ε cri and α should not be too small at the same time.

相较于现有技术，本发明具有以下有益效果：Compared with the prior art, the present invention has the following beneficial effects:

(1)基向量法综合了LRC法、Universal法和PSWL法的各自特点；保留了LRC法能够反映脉动风荷载分布和结构响应分布主要信息的优点，改善了Universal法通过工程师个人判断选择等效静力风荷载基向量的不足，增加了PSWL法没有考虑等效静力风荷载分布合理性和计算值不被高估的假设；(1) The basis vector method combines the respective characteristics of the LRC method, the Universal method and the PSWL method; it retains the advantage that the LRC method can reflect the main information of the fluctuating wind load distribution and structural response distribution, and improves the Universal method through the engineer's personal judgment. The insufficiency of the base vector of the static wind load increases the assumption that the PSWL method does not consider the rationality of the distribution of the equivalent static wind load and the calculated value is not overestimated;

(2)综合考虑了计算精度和等效静力风荷载分布的合理性，更合理的指导桥梁设计中风荷载的计算。(2) The calculation accuracy and the rationality of the equivalent static wind load distribution are comprehensively considered, and the calculation of the wind load in the bridge design is more reasonably guided.

附图说明Description of drawings

图1为本发明工作流程图。Fig. 1 is the working flow chart of the present invention.

图2为斜拉桥整体布置图(单位：m)。Figure 2 shows the overall layout of the cable-stayed bridge (unit: m).

图3为标准主梁断面(单位：cm)。Figure 3 is the standard main beam section (unit: cm).

图4为斜拉桥有限元模型。Figure 4 shows the finite element model of the cable-stayed bridge.

图5为作用在主梁结构上的风荷载。Figure 5 shows the wind loads acting on the main beam structure.

图6为斜拉桥主梁抖振位移响应RMS值。Figure 6 shows the RMS value of the buffeting displacement response of the main beam of the cable-stayed bridge.

图7为等效静力风荷载本征模态矩阵ΦLRC示意图(前5阶)。Figure 7 is a schematic diagram of the eigenmode matrix Φ LRC of the equivalent static wind load (the first 5 orders).

图8为多目标等效静力风荷载分布。Figure 8 shows the multi-objective equivalent static wind load distribution.

图9为抖振位移响应极值比较。Figure 9 is a comparison of the extreme values of the buffeting displacement response.

图10为多目标等效静力风荷载分布变化率。Figure 10 shows the distribution change rate of the multi-objective equivalent static wind load.

图中：图6(a)为抖振响应RMS值，(b)为主梁位置对应图；图7中的(a),(b),(c)分别为竖向，水平，扭转三个方向的本征模态矩阵示意图；图8中(a)为竖向等效静力风荷载分布图，(b)为水平等效静力风荷载分布图，(c)为扭转等效静力风荷载分布图；图9中(a)，(b)，(c)分别为竖向，水平，扭转三个方向的抖振位移响应极值；图10中(a)，(b)，(c)分别为竖向，水平，扭转三个方向的多目标等效静力风荷载变化率。In the figure: Figure 6(a) is the RMS value of the buffeting response, (b) is the corresponding diagram of the main beam position; (a), (b), (c) in Figure 7 are the vertical, horizontal and torsional three respectively Schematic diagram of the eigenmode matrix in the direction; Figure 8 (a) is the vertical equivalent static wind load distribution diagram, (b) is the horizontal equivalent static wind load distribution diagram, (c) is the torsional equivalent static force Wind load distribution diagram; (a), (b), (c) in Figure 9 are the extreme values of the buffeting displacement response in the vertical, horizontal and torsional directions; (a), (b), ( c) The rate of change of the multi-objective equivalent static wind load in the vertical, horizontal and torsional directions, respectively.

具体实施方式Detailed ways

下面结合附图，对本发明的技术方案进行具体说明。The technical solutions of the present invention will be described in detail below with reference to the accompanying drawings.

如图1所示，本发明一种桥梁多目标等效静力风荷载的计算方法，所述计算方法包括4个参数，分别为等效静力风荷载矩阵FLRC、本征模态矩阵ΦLRC、基于本征模态矩阵的等效静力风荷载基向量{ΦLRC}N×k和组合系数向量C。其中：As shown in FIG. 1 , a method for calculating the multi-objective equivalent static wind load of a bridge according to the present invention includes four parameters, namely, the equivalent static wind load matrix F LRC and the eigenmode matrix Φ LRC , the equivalent static wind load basis vector {Φ LRC } N×k based on the eigenmode matrix, and the combined coefficient vector C. in:

等效静力风荷载矩阵FLRC：依据荷载-响应之间的关系，以脉动风响应极值

Figure BDA0001886622100000051
为等效目标，通过LRC法获得等效静力风荷载列向量，Equivalent static wind load matrix F LRC : According to the load-response relationship, the extreme values of the fluctuating wind response
Figure BDA0001886622100000051
For the equivalent target, the column vector of the equivalent static wind load is obtained by the LRC method,

Figure BDA0001886622100000052
Figure BDA0001886622100000052

式中，

Figure BDA0001886622100000053
是以结构第i节点处的响应
Figure BDA0001886622100000054
为等效目标，按照LRC法计算出的等效静力风荷载列向量；
Figure BDA0001886622100000055
为脉动风荷载的RMS值列向量；
Figure BDA0001886622100000056
为脉动风荷载与响应
Figure BDA0001886622100000057
之间的荷载-响应相关系数列向量；符号“⊙”表示矩阵之间的运算为对应元素相乘；In the formula,
Figure BDA0001886622100000053
is the response at the ith node of the structure
Figure BDA0001886622100000054
is the equivalent target, the column vector of the equivalent static wind load calculated according to the LRC method;
Figure BDA0001886622100000055
is the column vector of the RMS value of the fluctuating wind load;
Figure BDA0001886622100000056
for fluctuating wind loads and responses
Figure BDA0001886622100000057
The column vector of the load-response correlation coefficient between ; the symbol "⊙" indicates that the operation between the matrices is the multiplication of the corresponding elements;

将获得的等效静力风荷载列向量组成N×m维的等效静力风荷载矩阵FLRC，

Figure BDA0001886622100000061
The obtained equivalent static wind load column vectors are formed into an N×m-dimensional equivalent static wind load matrix F LRC ,
Figure BDA0001886622100000061

本征模态矩阵ΦLRC：采用本征正交分解(POD)技术，对于离散数据结构称为奇异值分解(SVD)技术，对荷载矩阵FLRC进行分解，Eigenmode matrix Φ LRC : Using eigenorthogonal decomposition (POD) technology, which is called singular value decomposition (SVD) technology for discrete data structures, the load matrix F LRC is decomposed,

FLRC＝UΣVT (2)F LRC = UΣV T (2)

式中，U为与矩阵FLRC的行数相同的N×N维正交矩阵，Σ＝diag(λ1…λN)为N×m维非负对角矩阵，且λ1＞λ2＞…＞λN≥0，V为与矩阵FLRC的列数相同的m×m维正交矩阵；符号“T”表示矩阵的共轭转置，对于实矩阵为转置运算；In the formula, U is an N×N-dimensional orthogonal matrix with the same number of rows as the matrix F LRC , Σ=diag(λ 1 …λ N ) is an N×m-dimensional non-negative diagonal matrix, and λ 1 >λ 2 >...>λ N ≥ 0, V is an m×m-dimensional orthogonal matrix with the same number of columns as the matrix F LRC ; the symbol "T" represents the conjugate transpose of the matrix, and it is a transpose operation for a real matrix;

令ΦLRC＝U，Q＝ΣVT，则FLRC可转化为POD分解的标准形式，从而得到本征模态矩阵ΦLRC；Let Φ LRC =U, Q=ΣV T , then F LRC can be transformed into the standard form of POD decomposition, thereby obtaining the eigenmode matrix Φ LRC ;

FLRC＝ΦLRCQ (3)F LRC = Φ LRC Q (3)

式中,Q为对应的主坐标矩阵；In the formula, Q is the corresponding principal coordinate matrix;

基于本征模态矩阵的等效静力风荷载基向量{ΦLRC}N×k：以本征模态矩阵ΦLRC的前k阶列向量作为构建多目标等效静力风荷载的基本向量。Equivalent static wind load basis vector based on eigenmode matrix {Φ LRC } N×k : The first k-order column vector of eigenmode matrix Φ LRC is used as the basic vector for constructing multi-objective equivalent static wind load .

组合系数向量C：C为等效静力风荷载基向量的组合系数列向量，为k×1阶列向量。定义以C为未知数的函数f(C)，Combination coefficient vector C: C is the column vector of the combination coefficient of the base vector of the equivalent static wind load, which is a column vector of order k×1. Define the function f(C) with C as the unknown,

Figure BDA0001886622100000062
Figure BDA0001886622100000062

式中，||*||2表示二范数，min(*)表示求最小值。In the formula, ||*|| 2 represents the two-norm, and min(*) represents the minimum value.

本发明中，求得等效静力风荷载基本向量{ΦLRC}N×k后，以桥梁抖振响应极值为等效目标得In the present invention, after the basic vector of equivalent static wind load {Φ LRC } N×k is obtained, the extreme value of the buffeting response of the bridge is used as the equivalent target to obtain

Figure BDA0001886622100000063
Figure BDA0001886622100000063

式中，

Figure BDA0001886622100000064
为抖振响应极值列向量(不含平均风荷载响应)，其中σr为抖振响应r对应的RMS值列向量；I为响应r的影响函数矩阵；
Figure BDA0001886622100000065
为本征模态矩阵的前k阶列向量组成的等效静力风荷载基向量；C为等效静力风荷载基向量的组合系数列向量，为k×1阶列向量；In the formula,
Figure BDA0001886622100000064
is the column vector of the extreme value of the buffeting response (excluding the average wind load response), where σr is the column vector of the RMS value corresponding to the buffeting response r; I is the influence function matrix of the response r;
Figure BDA0001886622100000065
is the equivalent static wind load base vector composed of the first k-order column vectors of the eigenmode matrix; C is the combination coefficient column vector of the equivalent static wind load base vector, which is a k×1-order column vector;

然后将基向量组合系数C根据最小二乘法准则按照式(4)求得最优数值解。结构的多目标等效静力风荷载可在求得组合系数C后求得，Then the basis vector combination coefficient C is used to obtain the optimal numerical solution according to the formula (4) according to the least square method. The multi-objective equivalent static wind load of the structure can be obtained after obtaining the combination coefficient C,

Figure BDA0001886622100000071
Figure BDA0001886622100000071

式中，

Figure BDA0001886622100000072
为多目标等效静力风荷载列向量。In the formula,
Figure BDA0001886622100000072
is the column vector of the multi-objective equivalent static wind load.

本发明中，求得等效静力风荷载

Figure BDA0001886622100000073
后，计算等效静力风荷载精度，以桥梁关键部位处的响应作为评判目标。定义结构第i个关键部位处响应的计算误差为，In the present invention, the equivalent static wind load is obtained
Figure BDA0001886622100000073
Then, the accuracy of the equivalent static wind load is calculated, and the response at the key parts of the bridge is used as the evaluation target. Define the calculation error of the response at the i-th critical part of the structure as,

Figure BDA0001886622100000074
Figure BDA0001886622100000074

式中，εi为结构第i个关键位置处的抖振响应计算误差；

Figure BDA0001886622100000075
为按照随机抖振理论计算得到的结构第i个关键点处的抖振响应极值，简称为精确值；
Figure BDA0001886622100000076
为根据式(6)计算得到的结构第i个关键位置处的抖振响应极值，简称为计算值；εcri为计算误差控制值，根据工程经验确定，一般可取为10％；若结构关键部位处响应的计算误差均满足式(7)，则认为等效静力风荷载的计算精度满足要求。where ε i is the calculation error of the buffeting response at the ith key position of the structure;
Figure BDA0001886622100000075
is the extreme value of the chattering response at the ith key point of the structure calculated according to the random chattering theory, referred to as the exact value for short;
Figure BDA0001886622100000076
is the extreme value of the buffeting response at the ith key position of the structure calculated according to formula (6), referred to as the calculated value for short; If the calculation error of the response at the position meets the formula (7), it is considered that the calculation accuracy of the equivalent static wind load meets the requirements.

本发明中，等效静力风荷载在保证结构抖振响应计算精度的前提下，须验算其分布的合理性。因等效静力风荷载是根据结构的总体响应等效获得，而非局部响应。如果等效静力风荷载的分布出现剧烈突变，很可能在静力荷载组合计算中出现构件局部破坏或失稳的现象。为避免出现上述错误结果，必须保证等效静力风荷载分布的合理性。定义等效静力风荷载的均值

Figure BDA0001886622100000077
和标准差
Figure BDA0001886622100000078
分别为，In the present invention, the rationality of the distribution of the equivalent static wind load must be checked under the premise of ensuring the calculation accuracy of the buffeting response of the structure. Because the equivalent static wind load is obtained equivalently according to the overall response of the structure, not the local response. If the distribution of the equivalent static wind load has a sharp sudden change, it is very likely that the local failure or instability of the component will occur in the static load combination calculation. In order to avoid the above erroneous results, the rationality of the equivalent static wind load distribution must be guaranteed. Defining the mean value of the equivalent static wind load
Figure BDA0001886622100000077
and standard deviation
Figure BDA0001886622100000078
respectively,

Figure BDA0001886622100000079
Figure BDA0001886622100000079

Figure BDA00018866221000000710
Figure BDA00018866221000000710

若结构相邻两点处的等效静力风荷载分布满足关系式(10),则认为该节点处的荷载不存在突变，等效静力风荷载分布的合理性满足要求.If the distribution of the equivalent static wind load at two adjacent points of the structure satisfies the relation (10), it is considered that there is no sudden change in the load at the node, and the rationality of the distribution of the equivalent static wind load meets the requirements.

Figure BDA00018866221000000711
Figure BDA00018866221000000711

式中，α为经验系数，根据工程经验确定，一般可取为10％-20％。In the formula, α is the empirical coefficient, which is determined according to engineering experience, and is generally 10%-20%.

计算精度和等效静力风荷载分布的合理性须同时满足，有其一不满足时，则需返回重新计算等效静力风荷载，直至两者满足。需要注意的是，εcri和α的取值不应同时过小。The calculation accuracy and the rationality of the equivalent static wind load distribution must be satisfied at the same time. If one of them is not satisfied, it is necessary to return to recalculate the equivalent static wind load until both are satisfied. It should be noted that the values of ε cri and α should not be too small at the same time.

以下为本发明一具体实例，以东海大桥主航道桥为例，并配合附图，作详细说明如下，但本发明并不限于此。The following is a specific example of the present invention, taking the main channel bridge of the East China Sea Bridge as an example, and the accompanying drawings are described in detail as follows, but the present invention is not limited thereto.

参考图1至图10，以及表1-4，表1-4如下：Referring to Figures 1 to 10, and Tables 1-4, Tables 1-4 are as follows:

表1抖振计算主要参数Table 1 Main parameters of buffeting calculation

Figure BDA0001886622100000081
Figure BDA0001886622100000081

表2多目标等效静力风荷载计算误差(单位：％)Table 2 Multi-objective equivalent static wind load calculation error (unit: %)

Figure BDA0001886622100000082
Figure BDA0001886622100000082

表3多目标等效静力风荷载均方根Table 3 Multi-objective equivalent static wind load root mean square

Figure BDA0001886622100000083
Figure BDA0001886622100000083

表4多目标等效静力风荷载变化率最大值Table 4 Maximum value of multi-objective equivalent static wind load rate of change

Figure BDA0001886622100000084
Figure BDA0001886622100000084

例1：example 1:

东海大桥主航道桥连接上海与洋山深水港，为跨径组合73+132+420+132+73＝830m的独塔单索面斜拉桥，整体布置如图2所示。主梁采用钢-混叠合箱梁，梁宽33m，中心梁高4m，标准主梁断面如图3所示；倒Y型混凝土桥塔高148m。本例中采用的有限元程序为ANSYS，斜拉桥有限元模型如图4所示。The main channel bridge of Donghai Bridge connects Shanghai and Yangshan Deepwater Port. It is a single-pylon and single-cable-plane cable-stayed bridge with a span combination of 73+132+420+132+73=830m. The overall layout is shown in Figure 2. The main girder is made of steel-mixed box girder with a beam width of 33m and a central beam height of 4m. The section of the standard main beam is shown in Figure 3; the inverted Y-shaped concrete bridge tower is 148m high. The finite element program used in this example is ANSYS, and the finite element model of the cable-stayed bridge is shown in Figure 4.

经模态分析得到东海大桥主梁前20阶主要振型和频率，并作为抖振计算的基础。实际分析中作用在主梁上的风速主要考虑平均风速U、与平均风同方向的水平脉动风速u(t)和竖向脉动风速w(t)。作用在主梁上的风荷载主要考虑阻力D、升力L和升力矩M，对应方向的变形分别为p、h和α。各参数正方向表示如图5所示。The main mode shapes and frequencies of the first 20 orders of the main girder of the Donghai Bridge are obtained by modal analysis, which are used as the basis for buffeting calculation. In the actual analysis, the wind speed acting on the main beam mainly considers the average wind speed U, the horizontal pulsating wind speed u(t) and the vertical pulsating wind speed w(t) in the same direction as the average wind. The wind load acting on the main beam mainly considers the resistance D, the lift L and the lift moment M, and the deformations in the corresponding directions are p, h and α, respectively. The positive direction of each parameter is shown in Figure 5.

以斜拉桥有限元模型为基础，采用同时考虑自激力和抖振力的计算模型对斜拉桥主梁进行耦合抖振频域分析。自激力采用Scanlan提出的基于18个颤振导数的自激力表达式；抖振力采用Davenport考虑气动导纳的抖振力表达式，其中气动导纳采用Sears函数的Liepmann简化表达式。自激力和抖振力表达式中的气动力参数通过主梁节段模型风洞试验获得；颤振导数

Figure BDA0001886622100000091
和
Figure BDA0001886622100000092
通过节段模型测振风洞试验获得，颤振导数
Figure BDA0001886622100000093
和P1 *～P6 *通过准定常理论推导获得，静力三分力系数及其变化率通过节段模型测力试验获得，0°风攻角时的数据见表1。抖振计算中水平向脉动风谱Suu(n)选用Kaimal谱；竖向脉动风谱Sww(n)选用Lumley–Panofsky修正风谱；水平和竖向脉动风的交叉谱根据文献(SimiuE,Scanlan R H.Wind Effects on Structures(Third Edition)[M].JohnWiley&Sons,Inc.1996)选用；空间相关性采用《公路桥梁抗风设计规范》(JTG/T D60-01-2004)建议的形式。抖振计算中的其它主要参数见表1，以0°风攻角为例。考虑主梁前20阶振型，按照CQC组合方式，得主梁抖振位移响应RMS值，如图6所示。根据图6，分别取斜拉桥主跨跨中、主跨四分点、边跨-L2和边跨-R2跨中作为关键点，如图2所示。Node1、Node3、Node5分别指边跨-L2、主跨和边跨-R2跨中位置，Node2和Node4指主跨四分点位置。Based on the finite element model of the cable-stayed bridge, the coupled buffeting frequency domain analysis of the main beam of the cable-stayed bridge is carried out by using the calculation model that considers both the self-excited force and the buffeting force. The self-excited force adopts the self-excited force expression based on 18 flutter derivatives proposed by Scanlan; the buffeting force adopts the buffeting force expression of Davenport considering the aerodynamic admittance, and the aerodynamic admittance adopts the Liepmann simplified expression of the Sears function. The aerodynamic parameters in the self-excited and buffeting force expressions are obtained from the main beam segment model wind tunnel test; the flutter derivative
Figure BDA0001886622100000091
and
Figure BDA0001886622100000092
Obtained from segmental model vibrometric wind tunnel tests, the flutter derivative
Figure BDA0001886622100000093
and P 1 * ~ P 6 * are obtained by quasi-steady theoretical derivation, the static three-component force coefficient and its rate of change are obtained by segmental model force test, and the data at 0° wind attack angle are shown in Table 1. In the buffeting calculation, the horizontal pulsating wind spectrum S uu (n) uses the Kaimal spectrum; the vertical pulsating wind spectrum S ww (n) uses the Lumley–Panofsky modified wind spectrum; the cross spectrum of the horizontal and vertical pulsating wind is based on the literature (SimiuE, Scanlan R H.Wind Effects on Structures(Third Edition)[M].JohnWiley&Sons,Inc.1996); the spatial correlation adopts the form recommended in "Code for Design of Highway Bridges against Wind" (JTG/T D60-01-2004). Other main parameters in the buffeting calculation are shown in Table 1, taking 0° wind angle of attack as an example. Considering the first 20 modes of the main beam, according to the CQC combination method, the RMS value of the buffeting displacement response of the main beam is obtained, as shown in Figure 6. According to Figure 6, the mid-span of the main span, the quarter point of the main span, the mid-span of the side span-L2 and the mid-span of the side span-R2 of the cable-stayed bridge are taken as key points, as shown in Figure 2. Node1, Node3, and Node5 refer to the midspan positions of side span-L2, main span, and side span-R2, respectively, and Node2 and Node4 refer to the quarter point positions of the main span.

确定等效静力风荷载矩阵FLRC。取峰值因子g＝3.5，根据

Figure BDA0001886622100000094
获得主梁抖振位移响应极值对应的等效静力风荷载矩阵FLRC。Determine the equivalent static wind load matrix F LRC . Take the peak factor g = 3.5, according to
Figure BDA0001886622100000094
The equivalent static wind load matrix F LRC corresponding to the extreme value of the buffeting displacement response of the main beam is obtained.

确定本征模态矩阵ΦLRC。确定等效静力风荷载矩阵FLRC后，经SVD分解得到等效静力风荷载本征模态矩阵ΦLRC，其前5阶本征模态分布如图7所示。Determine the eigenmode matrix Φ LRC . After the equivalent static wind load matrix F LRC is determined, the equivalent static wind load eigenmode matrix Φ LRC is obtained by SVD decomposition, and the distribution of the first five eigenmodes is shown in Figure 7.

确定基于本征模态矩阵的等效静力风荷载基向量{ΦLRC}N×k。取本征模态矩阵ΦLRC前k阶列向量组成等效静力风荷载基向量。Determine the equivalent static wind load basis vector {Φ LRC } N×k based on the eigenmode matrix. The first k-order column vectors of the eigenmode matrix Φ LRC are taken to form the basis vector of the equivalent static wind load.

按发明内容所述的方法，可以分别得到基向量取k＝1、3、6、10、20时的多目标等效静力风荷载

Figure BDA0001886622100000101
的分布如图8所示。将图8中的多目标等效静力风荷载分别作用在斜拉桥主梁上，得抖振响应极值计算值，如图9所示。当k＝10、20时抖振响应计算误差很小，为表示清晰，图9中仅给出了精确抖振响应极值和k＝1、3、6时的计算值。According to the method described in the content of the invention, the multi-objective equivalent static wind loads can be obtained when the basis vectors are k=1, 3, 6, 10, and 20, respectively.
Figure BDA0001886622100000101
The distribution is shown in Figure 8. The multi-objective equivalent static wind loads in Fig. 8 are respectively applied to the main girder of the cable-stayed bridge, and the calculated value of the buffeting response extreme value is obtained, as shown in Fig. 9. When k = 10, 20, the calculation error of the chattering response is very small. For the sake of clarity, only the exact extreme value of the chattering response and the calculated values when k = 1, 3, and 6 are given in Figure 9.

按发明内容所述方法，计算多目标等效静力风荷载精度，当k＝1、3、6、10、20时，关键位置处的抖振响应计算误差如表2所示。当k＝6时，所有关键位置处的响应误差均不大于7％，其中主跨关键点处的计算误差均不大于5％，可满足工程设计的要求。According to the method described in the content of the invention, the multi-objective equivalent static wind load accuracy is calculated. When k=1, 3, 6, 10, and 20, the calculation error of the buffeting response at the key position is shown in Table 2. When k=6, the response errors at all key positions are not greater than 7%, and the calculation errors at the key points of the main span are not greater than 5%, which can meet the requirements of engineering design.

按本发明内容所述方法，在满足计算精度前提下，还须验算等效静力风荷载分布的合理性。由图8可知，随参与计算的基向量数量的增加，多目标等效静力风荷载分布的离散性逐渐增大。

Figure BDA0001886622100000102
的均方根如表3所示。为进一步评估等效静力风荷载分布的合理性，根据式(10)计算出不同位置处风荷载分布的变化率，如图10所示。各情况下等效静力风荷载分布的变化率最大值如表4所示。由图10可知，等效静力风荷载分布在边跨的变化率明显大于主跨。当k不大于6时，等效静力风荷载在竖向、水平和扭转方向的分布变化率均不大于9％，可以满足工程设计的需要。According to the method described in the content of the present invention, on the premise of satisfying the calculation accuracy, the rationality of the distribution of the equivalent static wind load must also be checked. It can be seen from Fig. 8 that with the increase of the number of basis vectors involved in the calculation, the discreteness of the multi-objective equivalent static wind load distribution gradually increases.
Figure BDA0001886622100000102
The root mean square 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 locations is calculated according to formula (10), as shown in Figure 10. The maximum value of the change rate of the equivalent static wind load distribution in each case is shown in Table 4. It can be seen from Fig. 10 that the variation rate of the equivalent static wind load distribution in the side span is significantly larger than that in the main span. When k is not more than 6, the distribution change rate of the equivalent static wind load in the vertical, horizontal and torsional directions is not more than 9%, which can meet the needs of engineering design.

以上是本发明的较佳实施例，凡依本发明技术方案所作的改变，所产生的功能作用未超出本发明技术方案的范围时，均属于本发明的保护范围。The above are the preferred embodiments of the present invention, all changes made according to the technical solutions of the present invention, when the resulting functional effects do not exceed the scope of the technical solutions of the present invention, belong to the protection scope of the present invention.

Claims (2)

1.一种基于POD技术的桥梁多目标等效静力风荷载计算方法，其特征在于：首先，计算等效静力风荷载矩阵FLRC、本征模态矩阵ΦLRC、基于本征模态矩阵的等效静力风荷载基向量{ΦLRC}N×k和组合系数列向量C；然后，依据计算获得的等效静力风荷载基本向量{ΦLRC}N×k和组合系数列向量C，求得多目标等效静力风荷载矩阵
Figure FDA0003579781150000011
最后，对多目标等效静力风荷载矩阵
Figure FDA0003579781150000012
的计算精度和分布合理性进行评估；若桥梁结构关键部位处响应的计算误差和等效静力风荷载分布的合理性同时满足条件，则认为所求的等效静力风荷载矩阵
Figure FDA0003579781150000013
满足工程设计要求；1. A multi-objective equivalent static wind load calculation method for bridges based on POD technology, characterized in that: first, calculate the equivalent static wind load matrix F LRC , the eigenmode matrix Φ LRC , based on the eigenmode The matrix equivalent static wind load base vector {Φ LRC } N×k and the combination coefficient column vector C; then, based on the calculated equivalent static wind load base vector {Φ LRC } N×k and the combination coefficient column vector C, find the multi-objective equivalent static wind load matrix
Figure FDA0003579781150000011
Finally, for the multi-objective equivalent static wind load matrix
Figure FDA0003579781150000012
If the calculation error of the response at the key parts of the bridge structure and the rationality of the equivalent static wind load distribution meet the conditions at the same time, the equivalent static wind load matrix is considered to be
Figure FDA0003579781150000013
meet engineering design requirements; 该方法具体实现步骤如下：The specific implementation steps of this method are as follows: 步骤S1、计算等效静力风荷载矩阵FLRC、本征模态矩阵ΦLRC、基于本征模态矩阵的等效静力风荷载基向量{ΦLRC}N×k和组合系数列向量C；其中，Step S1, calculate the equivalent static wind load matrix F LRC , the eigenmode matrix Φ LRC , the equivalent static wind load basis vector {Φ LRC } N×k and the combination coefficient column vector C based on the eigenmode matrix ;in, 等效静力风荷载矩阵FLRC：Equivalent static wind load matrix F LRC : 依据荷载-响应之间的关系，以脉动风响应极值
Figure FDA0003579781150000014
为等效目标，通过LRC法获得等效静力风荷载列向量，According to the load-response relationship, the extreme value of the pulsating wind response
Figure FDA0003579781150000014
For the equivalent target, the column vector of the equivalent static wind load is obtained by the LRC method,
Figure FDA0003579781150000015
Figure FDA0003579781150000015
 式中，
Figure FDA0003579781150000016
是以结构第i节点处的响应
Figure FDA0003579781150000017
为等效目标，按照LRC法计算出的等效静力风荷载列向量；
Figure FDA0003579781150000018
为脉动风荷载的RMS值列向量；
Figure FDA0003579781150000019
为脉动风荷载与响应
Figure FDA00035797811500000110
之间的荷载-响应相关系数列向量；符号“⊙”表示矩阵之间的运算为对应元素相乘；In the formula,
Figure FDA0003579781150000016
is the response at the ith node of the structure
Figure FDA0003579781150000017
is the equivalent target, the column vector of the equivalent static wind load calculated according to the LRC method;
Figure FDA0003579781150000018
is the column vector of the RMS value of the fluctuating wind load;
Figure FDA0003579781150000019
for fluctuating wind loads and responses
Figure FDA00035797811500000110
The column vector of the load-response correlation coefficient between ; the symbol "⊙" indicates that the operation between the matrices is the multiplication of the corresponding elements; 将获得的等效静力风荷载列向量组成N×m维的等效静力风荷载矩阵FLRC，
Figure FDA00035797811500000111
The obtained equivalent static wind load column vectors are formed into an N×m-dimensional equivalent static wind load matrix F LRC ,
Figure FDA00035797811500000111
 本征模态矩阵ΦLRC：The eigenmode matrix Φ LRC : 采用POD技术，对于离散数据结构称为奇异值分解技术，对等效静力风荷载矩阵FLRC进行分解，Using POD technology, which is called singular value decomposition technology for discrete data structures, the equivalent static wind load matrix F LRC is decomposed, FLRC＝UΣVT (2)F LRC = UΣV T (2) 式中，U为与矩阵FLRC的行数相同的N×N维正交矩阵，Σ＝diag(λ1…λN)为N×m维非负对角矩阵，且λ1＞λ2＞…＞λN≥0，V为与矩阵FLRC的列数相同的m×m维正交矩阵；T表示矩阵的共轭转置，对于实矩阵为转置运算；In the formula, U is an N×N-dimensional orthogonal matrix with the same number of rows as the matrix F LRC , Σ=diag(λ 1 …λ N ) is an N×m-dimensional non-negative diagonal matrix, and λ 1 >λ 2 >...>λ N ≥ 0, V is an m×m-dimensional orthogonal matrix with the same number of columns as the matrix F LRC ; T represents the conjugate transpose of the matrix, which is a transpose operation for real matrices; 令ΦLRC＝U，Q＝ΣVT，则FLRC转化为POD分解的标准形式，从而得到本征模态矩阵ΦLRC；Let Φ LRC =U, Q=ΣV T , then F LRC is converted into the standard form of POD decomposition, thereby obtaining the eigenmode matrix Φ LRC ; FLRC＝ΦLRCQ (3)F LRC = Φ LRC Q (3) 式中，Q为对应的主坐标矩阵；In the formula, Q is the corresponding principal coordinate matrix; 基于本征模态矩阵的等效静力风荷载基向量{ΦLRC}N×k：以本征模态矩阵ΦLRC的前k阶列向量作为构建多目标等效静力风荷载的基本向量；Equivalent static wind load basis vector based on eigenmode matrix {Φ LRC } N×k : The first k-order column vector of eigenmode matrix Φ LRC is used as the basic vector for constructing multi-objective equivalent static wind load ; 组合系数列向量C：Combined coefficient column vector C: C为等效静力风荷载基向量的组合系数列向量，为k×1阶列向量；定义以C为未知数的函数f(C)，C is the column vector of the combination coefficient of the base vector of the equivalent static wind load, which is a column vector of order k × 1; the function f(C) with C as the unknown is defined,
Figure FDA0003579781150000021
Figure FDA0003579781150000021
 式中，||*||2表示二范数，min(*)表示求最小值；In the formula, ||*|| 2 represents the second norm, and min(*) represents the minimum value; 步骤S2、依据计算获得的等效静力风荷载基本向量{ΦLRC}N×k，以桥梁抖振响应极值为等效目标，得Step S2, according to the basic vector of equivalent static wind load {Φ LRC } N×k obtained by calculation, and taking the extreme value of the buffeting response of the bridge as the equivalent target, we get:
Figure FDA0003579781150000022
Figure FDA0003579781150000022
 式中，
Figure FDA0003579781150000023
为抖振响应极值列向量，不含平均风荷载响应，其中σr为抖振响应r对应的RMS值列向量；I为响应r的影响函数矩阵；
Figure FDA0003579781150000024
为本征模态矩阵的前k阶列向量组成的等效静力风荷载基向量；C为等效静力风荷载基向量的组合系数列向量，为k×1阶列向量；In the formula,
Figure FDA0003579781150000023
is the column vector of the extreme value of the buffeting response, excluding the average wind load response, where σ r is the column vector of the RMS value corresponding to the buffeting response r; I is the influence function matrix of the response r;
Figure FDA0003579781150000024
is the equivalent static wind load base vector composed of the first k-order column vectors of the eigenmode matrix; C is the combination coefficient column vector of the equivalent static wind load base vector, which is a k×1-order column vector; 而后，将组合系数列向量C根据最小二乘法准则按照式(4)求得最优数值解；结构的多目标等效静力风荷载在求得组合系数列向量C后求得，Then, the optimal numerical solution is obtained by using the combination coefficient column vector C according to formula (4) according to the least square method; the multi-objective equivalent static wind load of the structure is obtained after the combination coefficient column vector C is obtained,
Figure FDA0003579781150000025
Figure FDA0003579781150000025
 式中，
Figure FDA0003579781150000026
为多目标等效静力风荷载列向量；In the formula,
Figure FDA0003579781150000026
is the column vector of the multi-objective equivalent static wind load; 步骤S3、依据求得的多目标等效静力风荷载矩阵
Figure FDA0003579781150000027
计算等效静力风荷载精度，以桥梁关键部位处的响应作为评判目标；定义结构第i个关键部位处响应的计算误差为，Step S3, based on the obtained multi-objective equivalent static wind load matrix
Figure FDA0003579781150000027
To calculate the equivalent static wind load accuracy, the response at the key part of the bridge is used as the evaluation target; the calculation error of the response at the i-th key part of the structure is defined as,
Figure FDA0003579781150000028
Figure FDA0003579781150000028
 式中，εi为结构第i个关键位置处的抖振响应计算误差；
Figure FDA0003579781150000029
为按照随机抖振理论计算得到的结构第i个关键点处的抖振响应极值；
Figure FDA0003579781150000031
为根据式(6)计算得到的结构第i个关键位置处的抖振响应极值；εcri为计算误差控制值，根据工程经验确定；若结构关键部位处响应的计算误差均满足式(7)，则认为等效静力风荷载的计算精度满足要求；where ε i is the calculation error of the buffeting response at the ith key position of the structure;
Figure FDA0003579781150000029
is the extreme value of the buffeting response at the ith key point of the structure calculated according to the random buffeting theory;
Figure FDA0003579781150000031
is the extreme value of the buffeting response at the ith key position of the structure calculated according to formula (6); ε cri is the control value of the calculation error, which is determined according to engineering experience; if the calculation error of the response at the key position of the structure all satisfies formula (7) ), the calculation accuracy of the equivalent static wind load is considered to meet the requirements; 步骤S3中，等效静力风荷载在保证计算精度的前提下，须验算其分布的合理性：In step S3, under the premise of ensuring the calculation accuracy of the equivalent static wind load, the rationality of its distribution must be checked: 定义等效静力风荷载的均值
Figure FDA0003579781150000032
和标准差
Figure FDA0003579781150000033
分别为Defining the mean value of the equivalent static wind load
Figure FDA0003579781150000032
and standard deviation
Figure FDA0003579781150000033
respectively
Figure FDA0003579781150000034
Figure FDA0003579781150000034
Figure FDA0003579781150000035
Figure FDA0003579781150000035
 若结构相邻两点处的等效静力风荷载分布满足关系式(10)，则认为该节点处的荷载不存在突变，等效静力风荷载分布的合理性满足要求；If the distribution of the equivalent static wind load at two adjacent points of the structure satisfies the relation (10), it is considered that there is no sudden change in the load at the node, and the rationality of the distribution of the equivalent static wind load meets the requirements;
Figure FDA0003579781150000036
Figure FDA0003579781150000036
 式中，α为经验系数，根据工程经验确定；In the formula, α is the empirical coefficient, which is determined according to engineering experience; 等效静力风荷载的计算精度和等效静力风荷载分布的合理性须同时满足，有其一不满足时，则需返回重新计算等效静力风荷载，直至两者满足。The calculation accuracy of the equivalent static wind load and the rationality of the equivalent static wind load distribution must be satisfied at the same time. 2.根据权利要求1所述的基于POD技术的桥梁多目标等效静力风荷载计算方法，其特征在于，所述εcri取10％，α取10％-20％。2 . The multi-objective equivalent static wind load calculation method for bridges based on POD technology according to claim 1 , wherein the ε cri is taken as 10%, and the α is taken as 10%-20%. 3 .
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