CN112507415B - Three-way earthquake-resistant design earthquake motion generation method combining orthogonalization and influence matrix method - Google Patents
Three-way earthquake-resistant design earthquake motion generation method combining orthogonalization and influence matrix method Download PDFInfo
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Abstract
The invention discloses a three-way earthquake-resistant design earthquake motion generation method combining orthogonalization and an influence matrix method, which can be used in the field of structural earthquake-resistant design and analysis. The method realizes perfect matching with a target spectrum by utilizing an influence matrix method, and introduces the orthogonalization of the gram-schmitt to realize zero correlation coefficient between any two earthquake motion components, namely, to meet the statistical independence of any two direction components. The method has the advantages of strict calculation process and high calculation efficiency, and can simultaneously realize high-precision matching of the three-dimensional earthquake 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 three-dimensional earthquake-resistant design earthquake motion generation method.
Background
In many seismic codes and standards, nonlinear dynamic time-course response analysis is required for important engineering structures. The reliability of the nonlinear dynamic time-course response analysis is greatly affected by the selected three-directional seismic time-course components. In order to meet engineering requirements, the earthquake motion time of the earthquake-proof design is required to be matched with the earthquake-proof design response spectrum. In addition, three-way seismic design schedules are also required to be statistically independent of each other to avoid cancellation or reinforcement effects between seismic components in different directions.
There are many methods or techniques currently used to generate seismic engineering seismic schedules, such as methods based on fourier transforms or time-frequency transforms of wavelet transforms, methods of overlaying specially constructed wavelet functions at specific moments in the initial seismic schedule, and empirical mode function decomposition methods based on hilbert-yellow transforms. The existing earthquake-resistant design earthquake motion time generation method mainly aims at matching a time-course reaction spectrum with an earthquake-resistant design spectrum, and cannot achieve statistical independence among earthquake time-course components in different directions while meeting the matching precision of earthquake time-courses in all directions. The method lacks theoretical basis in the aspect of meeting the statistical independence of time intervals in different directions, needs a large amount of repeated calculation, and is long in calculation time.
Disclosure of Invention
The invention aims to: aiming at the defect that the prior art can only meet the requirement of matching with a target spectrum on a unidirectional component and cannot meet the requirement of statistical independence between three-way earthquake time intervals, the invention provides a three-way 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-way earthquake-resistant design earthquake motion generating method combining orthogonalization and influence matrix method comprises the following steps:
(1) Expanding the initial seismic time course based on eigenfunctions in a calculated required frequency range [ f min,fmax ], horizontal and vertical target response spectra And/>Discrete at M frequency points, respectively, the initial earthquake motion time course comprises a first horizontal initial acceleration time course/>Second horizontal 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 initial acceleration time course of the first horizontal direction is performed by utilizing an influence matrix method Stepwise tuning to a horizontal target spectrum/>A matched first horizontal acceleration time interval a H1 (t);
(3) In the second horizontal direction H2, the initial acceleration time course of the second horizontal direction is carried out by coupling the Galame-Schmitt orthogonalization method to each iteration of the influence matrix method Stepwise tuning to a horizontal target spectrum/>A second horizontal acceleration time interval a H2 (t) that matches and is orthogonal to the first horizontal acceleration time interval a H1 (t);
(4) In the vertical direction V, using an influence matrix method of coupling gram-Schmidt orthogonalization to make the initial acceleration time course in the vertical direction Stepwise tuning to a vertical target spectrum/>And a vertical acceleration time interval a V (t) which is matched and respectively orthogonal to both the first horizontal acceleration time interval a H1 (t) and the second horizontal acceleration a H2 (t).
Further, the step (3) includes:
(a) Selecting a second horizontal initial acceleration time interval in a second horizontal direction H2 As an initial earthquake motion time course, the time course/>, is obtained by utilizing an influence matrix methodTo horizontal target spectrum/>Matching is carried out;
(b) The ith-1 second horizontal acceleration time course obtained for the ith-1 iteration The time course/>, is normalized by means of the gram-schmitt orthogonalization methodIs adjusted to be orthogonal to the first horizontal acceleration time interval A H1 (t);
(c) In the ith iteration, the ith-1 th second horizontal acceleration time course after orthogonalization processing is performed by using an influence matrix method And horizontal target spectrum/>Matching to obtain the i second horizontal 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 directionA second horizontal acceleration time interval a H2 (t) that matches and is orthogonal to the first horizontal acceleration time interval a H1 (t).
Further, after each orthogonalization, i.e. after step (b), introducing a scale factor to accelerate the time course of the orthogonalization-processed i-1 th second horizontal directionTime course/>, scaled to have the same mean square value as the first horizontal acceleration time course a H1 (t)Then in step (c) the scaled time course/>, is usedAnd horizontal target spectrum/>Matching is carried out.
Further, the step (4) includes:
(i) In the vertical direction V, selecting an initial acceleration time course in the vertical direction As an initial earthquake motion time course, utilizing an influence matrix method to perform initial acceleration time course/>To vertical target spectrum/>Matching is carried out;
(ii) For the i-1 th vertical acceleration time course obtained by the i-1 th iteration The time course/>, is normalized by means of the gram-schmitt orthogonalization methodAdjusted to be respectively consistent with the acceleration time course/>, in the first horizontal directionOrthogonal to the second horizontal acceleration time interval a H2 (t);
(iii) In the ith iteration, the (i-1) th acceleration time course after orthogonalization processing is performed by utilizing an influence matrix method Tuning and vertical target spectrum/>Matching to obtain the i-th 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 with the vertical directionAnd a vertical acceleration time interval a V (t) which is matched and respectively orthogonal to both the first horizontal acceleration a H1 (t) and the second horizontal acceleration a H2 (t).
The beneficial effects are that: compared with the prior art, the method utilizes the orthogonality requirement to replace the statistical independence requirement, and meets the two time courses of orthogonality mutually, so that the statistical independence requirement is accurately met. Specifically, the invention realizes perfect matching with the target spectrum by utilizing an influence matrix method, introduces the gram-schmitt orthogonalization to realize zero correlation coefficient between any two earthquake motion components, namely, satisfies the statistical independence of any two direction components. The method has the advantages of strict calculation process and high calculation efficiency, and can simultaneously realize high-precision matching of the three-dimensional earthquake 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 a three-way seismic design earthquake motion by combining orthogonalization with an influence matrix method according to an embodiment of the invention.
Detailed Description
The technical scheme of the invention is further described below with reference to the accompanying drawings.
Referring to fig. 1, a three-way earthquake-resistant design earthquake motion generation method combining orthogonalization and influence matrix method includes the steps of:
(1) Will initially vibrate time course And/>Based on eigenfunction expansion in the calculated required frequency range [ f min,fmax ], the horizontal and vertical target reaction spectra/>And/>Are discrete at M frequency points.
Specifically, a horizontal target reaction profileAnd a vertical target reaction profileAre each discrete at M frequency points, i.e., f 1＝fmin,f2＝fmin+1,…,fM＝fmax.fmax and f min are the upper and lower frequency limits, respectively, of the target design spectrum. Three-way seismic acceleration records of selected duration T are developed into a first horizontal initial acceleration time course/>, using eigenfunctions, respectively, as followsSecond horizontal initial acceleration time course/>And initial acceleration time course in vertical direction/>
The frequencies of the eigenfunctions corresponding to the serial numbers N max and N min are the upper frequency limit f max and the lower frequency limit f min of the target reaction spectrum respectively. Coefficients ofAnd/>Coefficient vectors corresponding to initial acceleration time interval eigenfunctions in the first horizontal direction, the second horizontal direction and the vertical direction are respectively formed. /(I)Is an eigenfunction, wherein T is more than or equal to 0 and less than or equal to T, and N min≤n≤Nmax.
(2) Selecting a first horizontal initial acceleration time interval in the first horizontal direction H1For initial earthquake motion time course, utilizing influence matrix method to make first horizontal direction initial acceleration time course/>Stepwise tuning to a horizontal target spectrum/>The matched first horizontal acceleration time interval a H1 (t).
In the embodiment of the invention, the influence matrix method is an improved influence matrix method for matching the high-low frequency alternation with the target spectrum so as to realize accurate matching with different types of reaction spectrums. For specific implementation, reference is made to chinese patent publication CN110069836a, which is not described in detail herein.
(3) In the second horizontal direction H2, the initial acceleration time course of the second horizontal direction is utilizedAs an initial seismic time course, the initial time course is obtained by coupling the glamer-schmitt orthogonalization method to each iteration of the influence matrix methodStepwise tuning to a horizontal target spectrum/>A second horizontal acceleration time interval a H2 (t) that matches and is orthogonal to the first horizontal acceleration time interval a H1 (t).
Since the statistical independence in each specification is measured by the cross-correlation coefficient between time intervals, when the two time intervals are absolutely orthogonal, the cross-correlation coefficient is equal to zero. The two time courses with zero cross-correlation coefficient are also in a mutually orthogonal relationship, so that the time courses can be adjusted to be in a pairwise orthogonal relationship by using an orthogonalization method, and all specification requirements are strictly met.
The specific implementation steps are as follows:
(3.1) obtaining the (i-1) th acceleration time course in the second horizontal direction through the (i-1) th iteration calculated by the influence matrix method The time course/>, can be determined by using the Galame-Schmitt orthogonalization method as followsIs adjusted to be orthogonal to the first horizontal acceleration time interval a H1 (t):
wherein, The second horizontal acceleration time course obtained by orthogonalization is the inner product operation.
(3.2) To ensure that the mean square values of the two horizontal seismic time-course components are equal, a scale factor is introduced after each orthogonalization:
Will orthogonalize (i-1) a second horizontal acceleration time course Time course/>, scaled to have the same mean square value as the first horizontal acceleration time course a H1 (t)
(3.3) In the ith iteration, the acceleration is time-course by using the influence matrix methodAnd target reaction Spectrum/>Matching is performed to make/>With the target spectrum/>The matching is tighter, and the influence matrix method is also realized by referring to Chinese patent CN 110069836A. The i-th acceleration time course can be obtained:
(3.4) repeating the steps (3.1) to (3.3) until the matching precision with the target reaction spectrum meets the requirement, and finally obtaining the target design spectrum in the horizontal direction A second horizontal acceleration time interval a H2 (t) that matches and is orthogonal to the first horizontal acceleration time interval a H1 (t).
(4) In the vertical direction V, selecting an initial acceleration time course in the vertical directionAs an initial earthquake motion time course, the initial time course/>, is obtained by using an influence matrix method of coupling gram-schmitt orthogonalizationStepwise tuning to a vertical target spectrum/>The matched vertical acceleration time interval a V (t) is orthogonal to the first horizontal acceleration time interval a H1 (t) and the second horizontal acceleration time interval a H2 (t), respectively.
The method specifically comprises the following steps:
(4.1) obtaining the (i-1) th acceleration time course in the vertical direction through the (i-1) th iteration calculated by the influence matrix method The time course/>, can be determined by using the Galame-Schmitt orthogonalization method as followsAdjusted to be orthogonal to two horizontal time courses a H1 (t) and a H2 (t), respectively:
wherein,
(4.2) In the ith iteration, the orthogonalized vertical acceleration time course is determined by an influence matrix methodTuning and targeting response profile/>Matching, and obtaining an ith vertical 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 finally obtaining the target reaction spectrum in the vertical direction A vertical-direction design earthquake motion time interval a V (t) which is matched and respectively orthogonal to the first horizontal-direction acceleration time interval a H1 (t) and the second horizontal-direction acceleration time interval a H2 (t):
(5) According to the three-way earthquake-proof design acceleration time courses A H1(t)、AH2 (t) and A V (t) obtained by the steps, the corresponding speed and displacement time courses can be obtained by using the following formulas:
Wherein V and D respectively represent a velocity time course and a displacement time course, subscripts H1, H1 and V respectively refer to a first horizontal direction, a second horizontal direction and a vertical direction, And/>The eigenfunctions for composing displacement, velocity and acceleration, the first derivative of the eigenfunctions and the second derivative of the eigenfunctions, respectively.
In order to more clearly understand the performance of the method according to the invention, it was verified by the following experiment. A set of three-way earthquake motion courses including two horizontal directions and two vertical directions is selected as an initial course, the duration of each course is t=30s, the time interval is 0.005s, and the total time count is 6001. The CENA UHS design spectrum is selected as the target spectrum, the calculated frequency range of the target spectrum is [0.2,100] Hz, and the total frequency point number is 270. The matching precision is set according to the specification requirements as follows:
(1) In the low frequency range of f min < f < 0.3Hz, the relative error |e (f) | max < 6%;
(2) The relative error |e (f) | max is less than 0.2% within the range that f is less than or equal to 0.6Hz and f max;
(3) Average relative error in the range of f min≤f≤fmax
The accuracy of matching the earthquake motion time courses in the first horizontal direction H1, the second horizontal direction H2, and the vertical direction V obtained by the method with the target design spectrum, and the correlation coefficients between the components in each direction are shown in table1, respectively.
TABLE 1 parameters for generating reaction spectra matching CENA-UHS
From the relative errors, the response spectrum of the generated time course is closely matched with the target spectrum. The strong motion duration of the H1, H2 and V components, which are determined according to the Alia intensity calculation, from 5% to 75% is 21.355s, 21.270s and 21.605s respectively, and the requirement of the strong motion duration is met. The correlation coefficient between the generated time records is basically zero and is far smaller than 0.16, and the requirements of the specifications on the correlation coefficient are strictly met.
The invention realizes the accurate matching of the earthquake motion time course reaction spectrum and the target design spectrum based on an improved influence matrix method; and the gram-schmitt orthogonalization method is used in each iteration of the influence matrix method, so that the correlation coefficient between any two earthquake motion components in any two directions is strictly zero, the earthquake motion components in three mutually orthogonal directions are ensured to be statistically independent, and the generated time interval meets the basic requirement of the current specification. The method utilizes an influence matrix method in any direction to realize high-precision matching of the reaction spectrum and the target design spectrum, and combines a gram-schmitt orthogonalization method to obtain three-directional earthquake motion components which are exactly matched with the target spectrum and are orthogonal to each other. The iteration process of the invention is monotonous and convergent, 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 more reliable earthquake-proof design and analysis results in engineering practice.
Claims (10)
1. The three-way earthquake-resistant design earthquake motion generating method combining orthogonalization and influence matrix method is characterized by comprising the following steps:
(1) Expanding the initial seismic time course based on eigenfunctions in a calculated required frequency range [ f min,fmax ], horizontal and vertical target response spectra And/>Discrete at M frequency points, respectively, the initial earthquake motion time course comprises a first horizontal initial acceleration time course/>Second horizontal initial acceleration time course/>And initial acceleration time course in vertical directionT represents time, T being a selected duration;
(2) In the first horizontal direction H1, the initial acceleration time course of the first horizontal direction is performed by utilizing an influence matrix method Stepwise tuning to a horizontal target spectrum/>A matched first horizontal acceleration time interval a H1 (t);
(3) In the second horizontal direction H2, the initial acceleration time course of the second horizontal direction is carried out by coupling the Galame-Schmitt orthogonalization method to each iteration of the influence matrix method Stepwise tuning to a horizontal target spectrum/>A second horizontal acceleration time interval a H2 (t) that matches and is orthogonal to the first horizontal acceleration time interval a H1 (t);
(4) In the vertical direction V, using an influence matrix method of coupling gram-Schmidt orthogonalization to make the initial acceleration time course in the vertical direction Stepwise tuning to a vertical target spectrum/>And a vertical acceleration time interval a V (t) which is matched and respectively orthogonal to both the first horizontal acceleration time interval a H1 (t) and the second horizontal acceleration a H2 (t).
2. The method for generating the three-dimensional earthquake-resistant design earthquake motion by combining orthogonalization and influencing matrix method according to claim 1, wherein the following formula is used for expanding the initial earthquake motion time course based on eigenfunctions in the calculated required frequency range [ f min,fmax ] in the step (1):
Wherein the frequencies of the eigenfunctions corresponding to the serial numbers N max and N min are respectively the upper limit f max and the lower limit f min of the frequency of the target reaction spectrum, and the coefficients are the same And/>The coefficient vectors corresponding to initial acceleration time interval eigenfunctions respectively forming a first horizontal direction, a second horizontal direction and a vertical direction,/>Is eigenfunction/>Is a second derivative of (c).
3. The method for generating three-way earthquake-resistant design earthquake motion by combining orthogonalization and influencing matrix method as set forth in claim 2, wherein the step (3) comprises:
(a) Selecting a second horizontal initial acceleration time interval in a second horizontal direction H2 As an initial earthquake motion time course, the time course/>, is obtained by utilizing an influence matrix methodTo horizontal target spectrum/>Matching is carried out;
(b) The ith-1 second horizontal acceleration time course obtained for the ith-1 iteration The time course/>, is normalized by means of the gram-schmitt orthogonalization methodIs adjusted to be orthogonal to the first horizontal acceleration time interval A H1 (t);
(c) In the ith iteration, the ith-1 th second horizontal acceleration time course after orthogonalization processing is performed by using an influence matrix method And horizontal target spectrum/>Matching to obtain the i second horizontal 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 directionA second horizontal acceleration time interval a H2 (t) that matches and is orthogonal to the first horizontal acceleration time interval a H1 (t).
4. A three-way seismic design ground vibration generation method combining orthogonalization and influencing matrix method according to claim 3, wherein the step (b) glam-schmitt orthogonalization method uses the following formula:
wherein, The inner product operation is shown.
5. The method for generating a three-way seismic design earthquake motion combining orthogonalization and influencing matrix method as recited in claim 3, wherein step (3) further comprises, between steps (b) and (c): after each orthogonalization, introducing a scale factor to make the (i-1) th second horizontal acceleration time course after orthogonalization treatmentTime course/>, scaled to have the same mean square value as the first horizontal acceleration time course a H1 (t)Utilizing the scaled time course/>, in said step (c)And horizontal target spectrum/>Matching is carried out.
6. The method for generating three-dimensional earthquake-resistant designed earthquake motion by combining orthogonalization and influence matrix method according to claim 5, wherein the scaling factor is calculated according to the following formula:
The inner product operation is shown.
7. The method for generating three-way earthquake-resistant design earthquake motion by combining orthogonalization and influencing matrix method as set forth in claim 2, wherein the step (4) comprises:
(i) In the vertical direction V, selecting an initial acceleration time course in the vertical direction As an initial earthquake motion time course, utilizing an influence matrix method to perform initial acceleration time course/>To vertical target spectrum/>Matching is carried out;
(ii) For the i-1 th vertical acceleration time course obtained by the i-1 th iteration The time course/>, is normalized by means of the gram-schmitt orthogonalization methodIs adjusted to be orthogonal to the first horizontal acceleration time interval A H1 (t) and the second horizontal acceleration time interval A H2 (t), respectively;
(iii) In the ith iteration, the orthogonalization-processed vertical acceleration time course is utilized by an influence matrix method Tuning and vertical target spectrum/>Matching to obtain the i-th 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 with the vertical directionAnd a vertical acceleration time interval a V (t) which is matched and respectively orthogonal to both the first horizontal acceleration a H1 (t) and the second horizontal acceleration a H2 (t).
8. The method for generating three-way earthquake-resistant designed earthquake motion by combining orthogonalization and influencing matrix method as recited in claim 7, wherein the glam-schmitt orthogonalization method in the step (ii) adopts the following calculation formula:
wherein, The inner product operation is shown.
9. The method for generating a three-way seismic designed earthquake motion combining orthogonalization and influencing matrix method of claim 1, wherein the method further comprises: and (3) obtaining the corresponding speed and displacement time courses according to the first horizontal acceleration time course A H1 (t), the second horizontal acceleration time course A H2 (t) and the vertical acceleration time course A V (t) obtained in the steps (1) - (4).
10. The method for generating three-way earthquake-resistant designed earthquake motion by combining orthogonalization and influencing matrix method as set forth in claim 9, wherein the speed and displacement time course is calculated as follows:
Wherein V and D respectively represent a velocity time course and a displacement time course, and subscripts H1, H2 and V respectively refer to a first horizontal direction, a second horizontal direction and a vertical direction.
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