CN112507415A - Three-dimensional earthquake-resistant design earthquake motion generation method combining orthogonalization and influence matrix method - Google Patents
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Abstract
The invention discloses a three-dimensional earthquake-resistant design earthquake motion generation method combining an orthogonalization and influence matrix method, which can be used for the field of structural earthquake-resistant design and analysis. The method realizes perfect matching with a target spectrum by using an influence matrix method, and introduces gram-Schmidt orthogonalization to realize that the correlation coefficient between any two seismic motion components is zero, namely the statistical independence of any two-direction components is met. The method is precise in calculation process and high in calculation efficiency, and can simultaneously realize high-precision matching of the three-dimensional seismic motion time course and the target design spectrum and statistical independence between every two time courses.
Description
Technical Field
The invention relates to a structural earthquake-resistant design and analysis method, in particular to a method for generating earthquake motion in a three-dimensional earthquake-resistant design.
Background
In many seismic codes and standards, nonlinear dynamic time-course response analysis is required to be performed on important engineering structures. The reliability of the nonlinear dynamic time-course response analysis is greatly influenced by the earthquake motion time-course components in the three selected directions. In order to meet engineering requirements, firstly, seismic design seismic motion time duration is required to be matched with a seismic design reaction spectrum. In addition, three-way seismic design schedules are required to be statistically independent from each other to avoid cancellation or strengthening effects between seismic components in different directions.
There are many methods or techniques currently used to generate seismic design seismic time-courses, such as a time-frequency domain transform method based on fourier transform or wavelet transform, a method of superimposing specially constructed wavelet functions at a specific time of an initial seismic time-course, and an empirical mode function decomposition method based on hilbert-yellow transform. The existing earthquake motion time-course generation method for earthquake design mainly aims at realizing matching of a time-course reaction spectrum and an earthquake design spectrum, and can not meet the requirement of the matching precision of earthquake time-courses in all directions and simultaneously realize the statistical independence among earthquake time-course components in different directions. The method is lack of theoretical basis in the aspect of meeting the statistical independence among time courses in different directions, needs a large amount of repeated calculation and is long in calculation time.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the defects that the prior art can only meet the matching with a target spectrum on a single directional component and cannot meet the requirement of three-dimensional earthquake time interval statistic independence, the invention provides a three-dimensional earthquake-resistant design earthquake motion generation method combining orthogonalization and an influence matrix method, and solves the problems.
The technical scheme is as follows: a three-dimensional earthquake-resistant design earthquake motion generation method combining an orthogonalization and influence matrix method comprises the following steps:
(1) setting the initial seismic motion time in the frequency range [ f ] required by calculationmin,fmax]Spread based intrinsic function internally, horizontal and vertical target response spectra
And
respectively dispersing at M frequency points, the initial seismic motion time course including a first horizontal directionInitial acceleration time course
Second horizontal direction initial acceleration time course
And initial acceleration time course in vertical direction
T represents time, T being a selected duration;
(2) in the first horizontal direction H1, the influence matrix method is used to make the time course of the initial acceleration in the first horizontal direction
Stepwise adjustment to the horizontal target spectrum
Matched first horizontal direction acceleration time course AH1(t)；
(3) In the second horizontal direction H2, coupling the gram-Schmidt orthogonalization method to each iteration of the influence matrix method, and time-ranging the initial acceleration of the second horizontal direction
Stepwise adjustment to the horizontal target spectrum
Matched with the first horizontal direction acceleration time course AH1(t) orthogonal second horizontal direction acceleration time course AH2(t)；
(4) In the vertical direction V, the time course of the initial acceleration in the vertical direction is processed by using a coupling gram-Schmidt orthogonalization influence matrix method
Gradually adjusted to the vertical target spectrum
Are matched with the first horizontal direction acceleration time course A respectivelyH1(t) and a second horizontal direction acceleration AH2(t) vertical acceleration time course AV(t)。
Further, the step (3) includes:
(a) selecting a second horizontal direction initial acceleration time interval in a second horizontal direction H2
As the initial seismic motion time course, the time course is processed by using an influence matrix method
Horizontal object spectrum
Matching is carried out;
(b) for the i-1 th second horizontal direction acceleration time course obtained by the i-1 st iteration
The time course is determined by using a gram-Schmidt orthogonalization method
Adjusted to the acceleration time course A in the first horizontal directionH1(t) orthogonal;
(c) in the ith iteration, the (i-1) th second horizontal direction acceleration time interval after the orthogonalization processing is carried out by utilizing an influence matrix method
And horizontal target spectra
Matching to obtain the ith second horizontal direction acceleration time course
(d) Repeating the steps (b) and (c) until the matching precision with the target reaction spectrum meets the requirement, and finally obtaining the target reaction spectrum in the horizontal direction
Matched with the first horizontal direction acceleration time course AH1(t) orthogonal second horizontal direction acceleration time course AH2(t)。
Further, after each orthogonalization, namely after the step (b), introducing a scale factor to orthogonalize the i-1 th second horizontal direction acceleration time interval after the orthogonalization
Scaled to a first horizontal acceleration time course AH1(t) time courses having the same mean square value
Then using the scaled time interval in step (c)
And horizontal target spectra
And matching is carried out.
Further, the step (4) includes:
(i) in the vertical direction V, selecting the initial acceleration time course in the vertical direction
As an initial seismic motion time interval, an influence matrix method is utilized to carry out an initial acceleration time interval in the vertical direction
To the vertical target spectrum
Matching is carried out;
(ii) for the i-1 th vertical direction acceleration time course obtained by the i-1 st iteration
The time course is determined by using a gram-Schmidt orthogonalization method
Adjusted to the acceleration time course in the first horizontal direction
And a second horizontal direction acceleration time course AH2(t) orthogonal;
(iii) in the ith iteration, the (i-1) th acceleration time interval in the vertical direction after the orthogonalization processing is carried out by utilizing an influence matrix method
Adjusting and vertical target spectra
Matching to obtain the ith vertical acceleration time course
(iv) Repeating the steps (ii) and (iii) until the matching precision with the target design spectrum meets the requirement, and finally obtaining the target reaction spectrum in the vertical direction
Are matched with the first horizontal direction acceleration A respectivelyH1(t) and a second horizontal direction acceleration AH2(t) acceleration time course A in vertical direction all orthogonal to each otherV(t)。
Has the advantages that: compared with the prior art, the method utilizes the orthogonality requirement to replace the statistic independence requirement, and meets two time courses of orthogonality mutually, thereby meeting the statistic independence requirement certainly and accurately. Specifically, the method realizes perfect matching with a target spectrum by using an influence matrix method, and introduces gram-Schmidt orthogonalization to realize that the correlation coefficient between any two seismic motion components is zero, namely the statistical independence of any two directional components is met. The method is precise in calculation process and high in calculation efficiency, and can simultaneously realize high-precision matching of the three-dimensional seismic motion time course and the target design spectrum and statistical independence between every two time courses.
Drawings
FIG. 1 is a flow chart of a method for generating earthquake motion in three-dimensional earthquake-resistant design by combining an orthogonalization and influence matrix method according to an embodiment of the invention.
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings.
Referring to fig. 1, a three-dimensional earthquake-resistant design earthquake motion generation method combining an orthogonalization and influence matrix method includes the following steps:
(1) moving the initial earthquake in time
And
in calculating the required frequency range fmin,fmax]Spread based on eigen function internally, and spectrum the horizontal and vertical target response
And
are all discrete at M frequency points.
In particular, horizontal target response profiles
And vertical target response spectra
Are respectively scattered at M frequency points, i.e. f1＝fmin,f2＝fmin+1,…,fM＝fmax。fmaxAnd fminThe upper and lower frequency limits of the spectrum are designed for the target, respectively. Expanding the three-way seismic acceleration record of the selected duration T into a first horizontal direction initial acceleration time course by utilizing an eigen function according to the following formula
Second horizontal direction initial acceleration time course
And initial acceleration time course in vertical direction
Wherein, the serial number NmaxAnd NminThe frequencies of the corresponding eigenfunctions are respectively the upper frequency limit f of the target reaction spectrummaxAnd a lower limit fmin. Coefficient of performance
And
and coefficient vectors corresponding to eigenfunctions forming the initial acceleration time course in the first horizontal direction, the second horizontal direction and the vertical direction respectively.
Is an eigenfunction, where T is greater than or equal to 0 and less than or equal to T, Nmin≤n≤Nmax。
(2) Selecting a first horizontal direction initial acceleration time interval in a first horizontal direction H1
For the initial earthquake motion time course, the first horizontal direction initial acceleration time course is processed by an influence matrix method
Stepwise adjustment to the horizontal target spectrum
Matched first horizontal direction acceleration time course AH1(t)。
In the embodiment of the invention, the influence matrix method is an improved influence matrix method for matching high-frequency and low-frequency alternation with the target spectrum so as to realize accurate matching with different types of reaction spectrums. The specific implementation method can refer to chinese patent publication No. CN110069836A, which is not described in detail herein.
(3) In the second horizontal direction H2, using the initial acceleration time course of the second horizontal direction
As an initial seismic motion time course, the initial time course is coupled to each iteration of the influence matrix method by a gram-Schmidt orthogonalization method
Stepwise adjustment to the horizontal target spectrum
Matched with the first horizontal direction acceleration time course AH1(t) orthogonal second horizontal direction acceleration time course AH2(t)。
Since the statistical independence in each specification is measured by the cross-correlation coefficient between the time spans, the cross-correlation coefficient is equal to zero when the two time spans are absolutely orthogonal. Two time intervals with zero cross correlation coefficient are also in a mutually orthogonal relationship, so that the time intervals can be adjusted to be in a pairwise orthogonal relationship by an orthogonalization method, and the requirements of each specification can be strictly met.
The specific implementation steps are as follows:
(3.1) obtaining an (i-1) th second horizontal direction acceleration time course through the (i-1) th iteration of the calculation of the influence matrix method
The time course can be determined by the following formula using the gram-schmitt orthogonalization method
Adjusted to the acceleration time course A in the first horizontal directionH1(t) orthogonal:
wherein the content of the first and second substances,
the time course of the second horizontal acceleration obtained by the orthogonalization process is an inner product operation.
(3.2) in order to ensure that the mean square values of the seismic motion time-course components in the two horizontal directions are equal, after each orthogonalization, introducing a scale factor:
(ii) the acceleration time course in the second horizontal direction after orthogonalizing (i-1) times
Scaled to a first horizontal acceleration time course AH1(t) time courses having the same mean square value
(3.3) in the ith iteration, using the influence matrix method to time-lapse the acceleration
And target reaction spectrum
Performing matching to
With the target spectrum
The matching is more compact, and the influence matrix method is realized by referring to the Chinese patent CN 110069836A. The ith acceleration time interval can be obtained:
(3.4) repeating the steps (3.1) - (3.3) until the matching precision of the target reaction spectrum meets the requirement, and finally obtaining the target design spectrum in the horizontal direction
Matched with the first horizontal direction acceleration time course AH1(t) orthogonal second horizontal direction acceleration time course AH2(t)。
(4) In the vertical direction V, selecting the initial acceleration time course in the vertical direction
As the initial seismic motion time course, the initial time course is processed by using a coupling gram-Schmidt orthogonalized influence matrix method
Gradually adjusted to the vertical target spectrum
Matched vertical acceleration time course AV(t) and the first horizontal acceleration time course AH1(t) and a second horizontal direction acceleration time course AH2(t) are orthogonal, respectively.
The method specifically comprises the following steps:
(4.1) obtaining the (i-1) th vertical direction acceleration time course through the (i-1) th iteration calculated by the influence matrix method
The time course can be determined by the following formula using the gram-schmitt orthogonalization method
Adjusted to two horizontal directionsH1(t) and AH2(t) are orthogonal respectively:
wherein the content of the first and second substances,
(4.2) in the ith iteration, utilizing an influence matrix method to orthogonalize the time course of the vertical direction acceleration
Adjustment of target response spectra
And matching to obtain the ith vertical direction acceleration time course:
(4.3) repeating the steps (4.1) and (4.2) until the matching precision with the target design spectrum meets the requirement, and finallyFinally obtaining a target reaction spectrum in the vertical direction
Matched with the first horizontal direction acceleration time course AH1(t) and a second horizontal direction acceleration time course AH2(t) designing earthquake motion time course A in vertical direction orthogonal to each otherV(t)：
(5) Obtaining three-dimensional anti-seismic design acceleration time course A according to the stepsH1(t)、AH2(t) and AV(t) obtaining corresponding velocity and displacement time course by using the following formulas:
wherein V and D respectively represent a speed time course and a displacement time course, lower corner marks H1, H1 and V respectively indicate a first horizontal direction, a second horizontal direction and a vertical direction,
and
respectively, an eigenfunction, a first derivative of the eigenfunction, and a second derivative of the eigenfunction for composing the displacement, velocity, and acceleration.
In order to more clearly understand the performance of the proposed method of the present invention, it was verified by the following experiments. A group of three-way earthquake motion time courses comprising two horizontal directions and two vertical directions are selected as initial time courses, the duration length of each time course is T-30 s, the time interval is 0.005s, and the total time point number is 6001. The CENA UHS design spectrum is selected as the target spectrum, the calculation frequency range of the target spectrum is [0.2,100] Hz, and the total frequency point number is 270. The matching accuracy is set according to the specification requirements as follows:
(1) at fminIn the low frequency range of < f < 0.3Hz, the relative error | e (f) & gtmax＜6％；
(2) F is more than or equal to 0.6Hz and less than fmaxWithin the range, the relative error | e (f) & gtdoes not countmax＜0.2％；
(3) At fmin≤f≤fmaxAverage relative error in range
The seismic motion time in the first horizontal direction H1, the second horizontal direction H2, the vertical direction V, the matching precision with the target design spectrum and the correlation coefficient between every two direction components obtained by the method are respectively shown in Table 1.
TABLE 1 parameters for generating a CENA-UHS matched reaction Profile
As can be seen from the relative error, the response spectrum of the generated time course closely matches the target spectrum. The duration of the strong movement of which the components H1, H2 and V are determined according to the calculation of the Aliasing intensity and are increased from 5% to 75% is 21.355s, 21.270s and 21.605s respectively, and the requirement of the duration of the strong movement is met. The correlation coefficient between the generated time records is basically zero and is far less than 0.16, and the requirement of the specification on the correlation coefficient is strictly met.
The method is based on an improved influence matrix method, and realizes the accurate matching of the seismic motion time-course response spectrum and the target design spectrum; in each iteration of the influence matrix method, a gram-Schmidt orthogonalization method is used, so that the correlation coefficient between seismic motion components in any two directions is strictly zero, the seismic motion components in three mutually orthogonal directions are ensured to be statistically independent, and the generated time course meets the basic requirement of the current specification. The method utilizes an influence matrix method in any direction to realize high-precision matching of a reaction spectrum and a target design spectrum, and combines a gram-Schmidt orthogonalization method to obtain three items of design seismic motion, namely, seismic motion components in three directions are accurately matched with the target spectrum and are pairwise orthogonal. The invention has monotonous and convergent iterative process, ensures the orthogonality among the earthquake motion components in three directions, and avoids mutual reinforcement or weakening among the components in different directions, thereby ensuring that the results of earthquake design and analysis in engineering practice are more reliable.
Claims (10)
1. The three-dimensional earthquake-resistant design earthquake motion generation method combining the orthogonalization and the influence matrix method is characterized by comprising the following steps of:
(1) setting the initial seismic motion time in the frequency range [ f ] required by calculationmin,fmax]Spread based intrinsic function internally, horizontal and vertical target response spectra
And
respectively dispersing at M frequency points, wherein the initial seismic motion time interval comprises a first horizontal direction initial acceleration time interval
Second horizontal direction initial acceleration time course
And initial acceleration time course in vertical direction
T represents time, T being a selected duration;
(2) in the first horizontal direction H1,time interval of initial acceleration in first horizontal direction by using influence matrix method
Stepwise adjustment to the horizontal target spectrum
Matched first horizontal direction acceleration time course AH1(t)；
(3) In the second horizontal direction H2, coupling the gram-Schmidt orthogonalization method to each iteration of the influence matrix method, and time-ranging the initial acceleration of the second horizontal direction
Stepwise adjustment to the horizontal target spectrum
Matched with the first horizontal direction acceleration time course AH1(t) orthogonal second horizontal direction acceleration time course AH2(t)；
(4) In the vertical direction V, the time course of the initial acceleration in the vertical direction is processed by using a coupling gram-Schmidt orthogonalization influence matrix method
Gradually adjusted to the vertical target spectrum
Are matched with the first horizontal direction acceleration time course A respectivelyH1(t) and a second horizontal direction acceleration AH2(t) vertical acceleration time course AV(t)。
2. The method for generating earthquake motion in three-dimensional earthquake-proof design combining orthogonalization and influence matrix method according to claim 1, wherein the step (1) comprises setting the initial earthquake motion time in the required frequency range [ f ] for calculationmin,fmax]The intrinsic-based expansion uses the following formula:
wherein, the serial number NmaxAnd NminThe frequencies of the corresponding eigenfunctions are respectively the upper frequency limit f of the target reaction spectrummaxAnd a lower limit fminCoefficient of
And
respectively forming coefficient vectors corresponding to eigenfunctions of initial acceleration time courses in a first horizontal direction, a second horizontal direction and a vertical direction,
is an eigenfunction.
3. The method of generating seismic motion for three-dimensional seismic design combining orthogonalization and influence matrix method according to claim 2, wherein the step (3) comprises:
(a) selecting a second horizontal direction initial acceleration time interval in a second horizontal direction H2
As the initial seismic motion time course, the time course is processed by using an influence matrix method
Horizontal object spectrum
Matching is carried out;
(b) for the i-1 th second horizontal direction acceleration time course obtained by the i-1 st iteration
The time course is determined by using a gram-Schmidt orthogonalization method
Adjusted to the acceleration time course A in the first horizontal directionH1(t) orthogonal;
(c) in the ith iteration, the (i-1) th second horizontal direction acceleration time interval after the orthogonalization processing is carried out by utilizing an influence matrix method
And horizontal target spectra
Matching to obtain the ith second horizontal direction acceleration time course
(d) Repeating the steps (b) and (c) until the matching precision with the target reaction spectrum meets the requirement, and finally obtaining the target reaction spectrum in the horizontal direction
Matched with the first horizontal direction acceleration time course AH1(t) orthogonal second horizontal direction acceleration time course AH2(t)。
4. A method of generating three-dimensional seismic design seismic motion combining orthogonalization and an influence matrix method according to claim 3, wherein the step (b) of gram-schmidt orthogonalization uses the following formula:
wherein the content of the first and second substances,
(x) is the inner product operation.
5. The method of generating seismic motion for three-dimensional seismic design combining orthogonalization and an influence matrix method according to claim 3, wherein the step (3) further comprises, between the steps (b) and (c): after each orthogonalization, introducing a scale factor to orthogonalize the i-1 th second horizontal direction acceleration time interval
Scaled to a first horizontal acceleration time course AH1(t) time courses having the same mean square value
Using the scaled time interval in step (c)
And horizontal target spectra
And matching is carried out.
6. The method for generating three-dimensional earthquake-resistant design earthquake motion by combining the orthogonalization and influence matrix method according to claim 5, wherein the scale factor is calculated according to the following formula:
(x) is the inner product operation.
7. The method of generating three-dimensional seismic design seismic motion combining orthogonalization and an influence matrix method according to claim 2, wherein the step (4) comprises:
(i) in the vertical direction V, selecting the initial acceleration time course in the vertical direction
As an initial seismic motion time interval, an influence matrix method is utilized to carry out an initial acceleration time interval in the vertical direction
To the vertical target spectrum
Matching is carried out;
(ii) for the i-1 th vertical direction acceleration time course obtained by the i-1 st iteration
The time course is determined by using a gram-Schmidt orthogonalization method
Adjusted to the acceleration time interval A in the first horizontal directionH1(t) and a second horizontal direction acceleration time course AH2(t) orthogonal;
(iii) in the ith iteration, the time course of the acceleration in the vertical direction after the orthogonalization processing is carried out by utilizing an influence matrix method
Adjusting and vertical target spectra
Matching to obtain the ith vertical acceleration time course
(iv) Repeating the steps (ii) and (iii) until the matching precision with the target design spectrum meets the requirement, and finally obtaining the target reaction spectrum in the vertical direction
Are matched with the first horizontal direction acceleration A respectivelyH1(t) and a second horizontal direction acceleration AH2(t) acceleration time course A in vertical direction all orthogonal to each otherV(t)。
8. The method for generating seismic motion in a three-dimensional earthquake-resistant design by combining an orthogonalization and an influence matrix method according to claim 7, wherein the gram-schmidt orthogonalization in the step (ii) is calculated by using the following formula:
wherein the content of the first and second substances,
(x) is the inner product operation.
9. The method of generating three-dimensional seismic design seismic motions with a combination of orthogonalization and influence matrix method according to claim 1, further comprising: obtaining a first horizontal direction acceleration time course A according to the steps (1) to (4)H1(t) second horizontal acceleration time course AH2(t) and vertical acceleration time course AVAnd (t) obtaining the corresponding speed and displacement time course.
10. The method of generating three-dimensional seismic design seismic motion according to claim 9, wherein the velocity and displacement time interval are calculated as follows:
where V and D represent a speed time course and a displacement time course, respectively, and lower corner marks H1, H1, and V indicate a first horizontal direction, a second horizontal direction, and a vertical direction, respectively.
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