WO2023229116A1 - Method for calculating time-history wind loads in considertion of correlation - Google Patents
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Definitions
- the present invention is a time history wind load calculation method considering correlation that can generate an artificial time history load close to reality without relying on wind tunnel experiments by accurately reproducing the desired level of correlation by directly using the phase difference of the time history for two-way wind load. It's about.
- time history analysis of wind loads is necessary.
- Methods for generating time history loads for this purpose include methods using autoregressive analysis (Iwatani (1996), Hwang Jong-guk et al. (1998), etc.) and spectral expression methods (Hwang Jae-seung et al. (2015), etc.).
- f is the frequency
- ⁇ f is the frequency interval
- ⁇ i is a random phase angle that is a random value between 0 and 2 ⁇ .
- the power spectrum density function of the load in each direction is calculated through Fourier transform of the autocorrelation function, and information about the moment of maximum value is missing for the load in each direction.
- the coherence function which is widely used as an indicator to evaluate correlation by quantitatively displaying the linear correlation between two random processes, includes the ratio of the moment when the maximum value of one wind load occurs to the maximum value of the other wind load; In other words, it is difficult to predict the maximum load ratio.
- Coherence calculated through ensemble average becomes 1 when the phase difference and amplitude ratio between ensemble samples at specific frequency components of two time histories remain the same. In other words, coherence is only an indirect indicator of how well the ratio of phase difference and amplitude is maintained between ensembles.
- Registered Patent No. 10-2354932 reflects the correlation between wind loads in two directions and adjusts the ratio to the maximum value of the other wind load when one wind load is the maximum value, thereby creating an artificial artificial intelligence that is close to reality without relying on wind tunnel experiments. Time history loads can be created.
- the above method estimates correlation by roughly adjusting the coefficients a and b from coherence, an engineer's intervention is required, and a lot of trial and error is required to reach the desired level of correlation for each frequency.
- the present invention according to a preferred embodiment is provided to a computing device to reproduce the time history wind load for the first wind load, which is one of the wind direction load, the wind angle load, and the torsional load, and the second wind load, which is one of the remaining loads.
- step (a) is the ensemble average of the cosine value of the phase difference from the wind tunnel experiment data ( ) Data is provided and the ensemble average of the phase difference is calculated by the ensemble average of the cosine value of the phase difference ( ) provides a time history wind load calculation method considering correlation, which is characterized by assuming the equation below.
- step (b) is performed by calculating the ensemble standard deviation ( ) provides a time history wind load calculation method considering correlation, which is characterized by calculating ).
- time history wind load calculation considering correlation can generate an artificial time history load close to reality without relying on wind tunnel experiments by accurately reproducing the desired level of correlation by directly using the phase difference of the time history for two-way wind load.
- a method can be provided.
- the vibration mode of the structure can be more accurately reflected by calculating the response of the structure through accurate time history analysis, and nonlinear analysis in which the equivalent static load cannot be performed can be performed. You can.
- FIG. 1 is a diagram showing types of wind load acting on a structure.
- Figure 2 is a graph showing the phase difference defined by the relationship between the real part and the imaginary part of the CPSD function.
- Figure 4 is a graph showing the relationship between the phase difference and the ensemble standard deviation of the cosine of the phase difference.
- FIG. 5 is a flowchart showing a procedure for reproducing a correlated time history from phase difference information.
- Figure 6 is a graph showing the verification results of the average of the reproduced phase difference cosine.
- Figure 7 is a graph showing the verification results of the standard deviation of the reproduced phase difference cosine.
- Figure 8 is a graph showing the comparison results of wind tunnel experiment results and reproduced phase difference samples.
- the time history wind load calculation method considering the correlation of the present invention is the first wind load, which is one of the wind direction load, wind angle direction load, and torsional load, and the second wind load, which is one of the remaining loads. is performed by a computing device to reproduce the time history wind load for (a) the ensemble average of the phase difference between a random first random phase angle for the first wind load and a second random phase angle for the second wind load ( ) assuming that; (b) the ensemble average ( ) by the ensemble standard deviation ( ) calculating; (c) ensemble standard deviation of the phase difference ( ) by the ensemble average for the phase difference ( ) correcting; (d) the ensemble mean of the corrected phase difference and the ensemble standard deviation of the phase difference ( ), the phase difference between the first time history wind load and the second time history wind load at each frequency ( ) calculating; (e) Random first phase angle set with uniform distribution for the first time history ( ) to reproduce; (f) phase difference calculated in step (d) above ( ) and the random
- Figure 2 is a graph showing the phase difference defined by the relationship between the real part and the imaginary part of the CPSD function.
- the time history wind load calculation method considering the correlation of the present invention is to reproduce the time history wind load for the first wind load, which is one of the wind direction load, wind angle direction load, and torsional load, and the second wind load, which is one of the remaining loads.
- a computing device Performed by a computing device, (a) an ensemble average of the phase difference between a random first random phase angle for the first wind load and a second random phase angle for the second wind load ( ) assuming that; (b) the ensemble average ( ) by the ensemble standard deviation ( ) calculating; (c) ensemble standard deviation of the phase difference ( ) by the ensemble average for the phase difference ( ) correcting; (d) the ensemble mean of the corrected phase difference and the ensemble standard deviation of the phase difference ( ), the phase difference between the first time history wind load and the second time history wind load at each frequency ( ) calculating; (e) Random first phase angle set with uniform distribution for the first time history ( ) to reproduce; (f) phase difference calculated in step (d) above ( ) and the random
- the present invention accurately reproduces the desired level of correlation by directly using the phase difference of the time history for the two-way wind load, and calculates the time history wind load considering the correlation that can generate an artificial time history load close to reality without relying on wind tunnel experiments. It is intended to provide a method.
- the present invention is performed by a computing device to reproduce a time history wind load for a first wind load, which is one of a wind direction load, a wind angle load, and a twisting direction load, and a second wind load, which is one of the remaining loads.
- phase difference between the two time histories can be extracted from the Cross Power Spectral Density (CPSD) function of the wind tunnel experiment data.
- CPSD Cross Power Spectral Density
- the CPSD function is a complex number as shown in [Equation 3] below.
- phase difference can be defined by the relationship between the real part and the imaginary part of the CPSD function, and by [Equation 4], the phase difference can be expressed as [Equation 5].
- the phase difference information calculated in this way has a value between - ⁇ and ⁇ .
- the ensemble average ( ) and ensemble standard deviation ( ) is used.
- the ensemble average ( ) alone is not only difficult to properly identify trends, but also the ensemble average ( ) and ensemble standard deviation ( ) in the process of converting phase difference information, the average and standard deviation affect each other.
- step (a) the ensemble average for the phase difference ( ) is first assumed as the initial setting value.
- step (a) the ensemble average of the phase difference between the random first random phase angle for the first wind load and the second random phase angle for the second wind load ( ) is assumed.
- Steps (b) and (c) may be repeated as many times as necessary.
- step (d) the ensemble average of the phase difference corrected in step (c) and the ensemble standard deviation of the phase difference ( ), the phase difference between the first time history wind load and the second time history wind load, which are two time history wind loads at each frequency ( ) is calculated.
- the ensemble mean and ensemble standard deviation By generating random numbers that follow a normal distribution by ), the phase difference set ( ) is created.
- step (e) a random first phase angle set ( ) play.
- step (f) the first phase angle set generated in step (e) ( ) and a second phase angle set by the phase difference set generated in step (d) ( ) is calculated.
- the first time history wind load (X(t)) and the second time history wind load (Y(t)) are respectively shown in [Equation 6] and [Equation 7] below.
- an accurate time history load can be generated by utilizing the power spectrum density function and phase difference information established by many researchers without always relying on wind tunnel experiments.
- the vibration mode of the structure can be reflected more accurately by calculating the response of the structure through accurate time history analysis.
- nonlinear analysis can be performed where equivalent static loads cannot be used.
- the step (a) is the ensemble average of the cosine value of the phase difference from the wind tunnel experiment data ( ) Data is provided and the ensemble average of the phase difference is calculated by the ensemble average of the cosine value of the phase difference ( ) can be assumed by the [mathematical formula] below.
- phase difference information can be presented in the form of a cosine function.
- the cosine function By applying the cosine function, distortion due to the above periodic properties can be prevented.
- the ensemble average data for the cosine value of the phase difference is utilized.
- step (a) above the ensemble average ( )
- the initial value for the assumption can be expressed as [Equation 8] below.
- the sign is set so that the ensemble average value of the cosine of each other's phase differences is satisfied.
- Figure 3 is a graph showing the relationship between the phase difference and the ensemble average of the cosine of the phase difference.
- Step (c) above is performed by the ensemble average ( ) can be corrected.
- the value of the inverse cosine in order for the value of the inverse cosine to be defined as a real number, must have a value between -1 and 1, and if it exceeds this value, the value must be limited to -1 or 1.
- the average phase difference can be calculated as the initial setting value as in [Equation 8] and then corrected through a minimum repetition process of 0 or 1 time.
- Figure 4 is a graph showing the relationship between the phase difference and the ensemble standard deviation of the cosine of the phase difference.
- step (b) the ensemble standard deviation ( ) can be calculated.
- phase difference information is expressed as a cosine function
- the ensemble average ( ) alone cannot properly represent the tendency. If there is no correlation, the phase difference appears uniformly distributed between - ⁇ and ⁇ , and the ensemble average of the cosine of the phase difference is 0.
- phase difference ensemble will have a Gaussian distribution with a specific mean value and standard deviation.
- the results of the wind tunnel experiment are given in the form of the ensemble mean and ensemble standard deviation of the cosine of the phase difference, the relationship between them must be defined in order to restore the phase difference information from them.
- the present invention creates 10,000 phase differences and introduces an abbreviation equation that defines the relationship.
- Figure 5 is a flowchart showing a procedure for reproducing a correlated time history from phase difference information.
- Figure 5 shows a flowchart showing a time history wind load calculation method considering the correlation of the present invention.
- Time history wind loads X(t) and Y(t) can be obtained from phase difference information through the procedure shown in FIG. 5.
- Figures 6 to 8 are graphs showing the results of restoration and verification of phase difference information.
- Figure 6 shows the verification result of the average of the reproduced phase difference cosine
- Figure 7 shows the verification result of the standard deviation of the reproduced phase difference cosine.
- N represents the number of repetitions.
- Figure 8 shows the comparison results of the wind tunnel test results and the reproduced phase difference samples.
- the time history wind load calculation method considering the correlation of the present invention can generate an artificial time history load close to reality without relying on wind tunnel experiments by accurately reproducing the desired level of correlation by directly using the phase difference of the time history for two-way wind load. There is potential for industrial use in that it exists.
Abstract
The present invention relates to a method for calculating time-history wind loads in consideration of correlation, the method directly using time-history phase differences for bidirectional wind loads to accurately regenerate the desired level of correlation, thereby allowing artificial time-history wind loads to be generated that are close to reality without having to rely on wind tunnel tests.
Description
본 발명은 두 방향 풍하중에 대한 시간이력의 위상차를 직접 이용하여 원하는 수준의 상관성을 정확하게 재생함으로써 풍동실험에 의존하지 않고 실제에 가까운 인공 시간이력하중을 생성할 수 있는 상관성을 고려한 시간이력풍하중 산정 방법에 대한 것이다.The present invention is a time history wind load calculation method considering correlation that can generate an artificial time history load close to reality without relying on wind tunnel experiments by accurately reproducing the desired level of correlation by directly using the phase difference of the time history for two-way wind load. It's about.
시간에 따라 변화하는 풍하중을 건축물 설계에 고려하기 위해서는 풍하중에 대한 시간이력해석(time history analysis)이 필요하다. 이를 위한 시간이력하중 생성 방법으로는 크게 자기회귀분석을 이용한 방법(Iwatani(1996), 황종국 외 1인(1998) 등)과 스펙트럼 표현법(황재승 외 2인(2015) 등)이 있다.In order to consider wind loads that change with time in building design, time history analysis of wind loads is necessary. Methods for generating time history loads for this purpose include methods using autoregressive analysis (Iwatani (1996), Hwang Jong-guk et al. (1998), etc.) and spectral expression methods (Hwang Jae-seung et al. (2015), etc.).
실제 건축물에는 도 1과 같이 바람이 불어오는 풍방향 풍하중(W1), 바람 방향과 직교하는 풍직각 방향 풍하중(W2) 및 비틀림 풍하중(W3)의 하중이 동시에 작용한다. 그러므로 시간이력풍하중 생성 시 이러한 다양한 종류의 풍하중을 고려해야 한다.As shown in Figure 1, in an actual building, the wind load in the wind direction (W 1 ), the wind load in the direction perpendicular to the wind direction (W 2 ), and the torsional wind load (W 3 ) act simultaneously. Therefore, these various types of wind loads must be considered when generating time history wind loads.
건축구조기준 역시 여러 종류의 풍하중을 설계에 반영하기 위해, 각 방향 하중의 파워스펙트럼밀도(Power Spectral Density, PSD) 함수를 이용한 주파수 영역 해석을 통해 각 하중의 최댓값을 등가정적하중 형태로 제시하고 있다. 여기에서는 각 방향별로 하중의 최댓값을 산정하기는 하지만 실제로 각 방향의 하중 최댓값이 동시에 발생할 가능성은 거의 없다. 따라서 이를 보정하기 위해 각 방향별 하중의 최댓값에 하중조합계수를 적용하여 설계한다. 이때, 등가정적하중을 산정하는 과정에서 구조물의 공진 효과를 반영하기 위해 1차 모드 진동 형상을 가정하며, 고차 모드의 영향은 무시한다. 이에 고차 모드의 영향이 커지는 고층 건물에서는 등가정적하중의 정확성이 떨어진다. In order to reflect various types of wind loads in the design, building structural standards also present the maximum value of each load in the form of equivalent static load through frequency domain analysis using the Power Spectral Density (PSD) function of the load in each direction. . Here, the maximum value of the load in each direction is calculated, but in reality, it is unlikely that the maximum value of the load in each direction will occur simultaneously. Therefore, to correct this, the design is designed by applying the load combination coefficient to the maximum value of the load in each direction. At this time, in the process of calculating the equivalent static load, the first-order mode vibration shape is assumed to reflect the resonance effect of the structure, and the influence of higher-order modes is ignored. Accordingly, the accuracy of the equivalent static load is reduced in high-rise buildings where the influence of higher order modes increases.
또한, 고층 건물은 중력하중에 의한 비선형적인 P-델타 효과(Geometric Nonlinearity)의 영향이 크므로, 선형 중첩이 불가능하여 정확한 풍하중 생성에 한계가 있다. 뿐만 아니라 항복에 의해 비탄성 거동하는 비탄성 구조물의 경우에도 중첩이 불가능하므로, 모든 하중을 동시에 적용시키고 비선형 시간이력해석을 통해 응답을 구해야 한다.In addition, high-rise buildings are greatly affected by the nonlinear P-delta effect (Geometric Nonlinearity) caused by gravity loads, so linear overlap is impossible, which limits the ability to accurately generate wind loads. In addition, since superimposition is not possible in the case of inelastic structures that behave inelasticly due to yielding, all loads must be applied simultaneously and the response must be obtained through nonlinear time history analysis.
상기 등가정적하중을 이용한 설계의 한계를 극복하기 위해 풍동실험을 통해 측정된 시간이력하중을 사용하거나 파워스펙트럼밀도 함수로부터 인공 시간이력하중을 재생하여 사용할 수 있다. 그러나 풍동실험은 큰 비용과 시간이 소요되므로, 정형적인 형상의 건물은 이미 수행되어 많은 자료가 축적된 과거의 풍동실험 결과들로부터 정립된 파워스펙트럼밀도 함수를 이용하여 인공 시간이력하중을 재생하는 방법이 활용되기도 한다.In order to overcome the limitations of the design using the equivalent static load, the time history load measured through wind tunnel experiment can be used, or the artificial time history load can be reproduced from the power spectrum density function. However, since wind tunnel experiments require a large amount of cost and time, a method of reproducing artificial time history loads using the power spectrum density function established from the results of past wind tunnel experiments that have already been performed on buildings with regular shapes and a lot of data has been accumulated This is also used.
일방향 파워스펙트럼밀도(One-Sided PSD) 함수 S(f)가 주어질 경우, 시간이력하중 X(t)는 다음 [수학식 1] 과 같이 재생할 수 있다.When the one-sided power spectrum density (One-Sided PSD) function S(f) is given, the time history load X(t) can be reproduced as follows [Equation 1].
[수학식 1][Equation 1]
여기서, f는 주파수, Δf는 주파수 간격, θi는 0~2π의 임의의 값인 랜덤 위상각을 의미한다.Here, f is the frequency, Δf is the frequency interval, and θ i is a random phase angle that is a random value between 0 and 2π.
이때, 각 방향 하중의 파워스펙트럼밀도 함수는 자기상관함수의 푸리에 변환을 통해 산정되며, 방향별 하중에 대해서는 최댓값 순간에 대한 정보가 누락된다. At this time, the power spectrum density function of the load in each direction is calculated through Fourier transform of the autocorrelation function, and information about the moment of maximum value is missing for the load in each direction.
또한, 각 방향 하중 재생 시 서로 독립적인 위상각 θ를 적용할 경우, 각 방향의 풍하중들 간 상관성이 반영되지 않는다. 반대로 각 방향의 풍하중들에 대해 완전히 동일한 위상각 θ를 적용할 경우, 상관성이 과도하게 되고 각 방향 하중들의 최댓값이 동시에 발생할 가능성이 과도하게 커지는 문제가 있다. 아울러 두 랜덤 프로세스의 선형적인 상호 관계를 양적으로 표시하여 상관성을 평가하는 지표로 많이 활용되는 코히어런스 함수(coherence function)로는 어느 한 풍하중의 최댓값이 발생하는 순간에서 다른 풍하중의 최댓값에 대한 비율, 즉 최대하중비를 예측하기 어렵다.Additionally, when independent phase angles θ are applied when reproducing loads in each direction, the correlation between wind loads in each direction is not reflected. Conversely, if the completely same phase angle θ is applied to the wind loads in each direction, there is a problem that the correlation becomes excessive and the possibility that the maximum values of the loads in each direction occur simultaneously increases excessively. In addition, the coherence function, which is widely used as an indicator to evaluate correlation by quantitatively displaying the linear correlation between two random processes, includes the ratio of the moment when the maximum value of one wind load occurs to the maximum value of the other wind load; In other words, it is difficult to predict the maximum load ratio.
각 방향별 풍하중의 상관성을 반영하기 위해 기존에는 주파수별 상관을 나타내는 지표로 크로스파워스펙트럼밀도(Cross Power Spectral Density, CPSD) 함수를 PSD 함수들로 일반화한 아래 [수학식 2] 와 같은 코히어런스 함수를 사용한다.In order to reflect the correlation of wind load in each direction, the Cross Power Spectral Density (CPSD) function, which was previously an indicator of correlation by frequency, was generalized to PSD functions, and the coherence as shown in [Equation 2] below Use functions.
[수학식 2][Equation 2]
여기서, SX(f), SY(f)는 주파수에 대한 파워스펙트럼밀도 함수, SXY(f)는 크로스파워스펙트럼밀도 함수이다. Here , S
코히어런스는 하나의 샘플에서 산정할 경우 그 값이 항상 1이 된다. 그렇기 때문에, 0~1 사이에서 1 이외의 값을 갖기 위해서는 각 성분의 앙상블 평균으로 산정하거나 각 주파수 구간을 나누고 이동평균 형태로 산정해야 한다.When coherence is calculated from one sample, its value is always 1. Therefore, in order to have a value other than 1 between 0 and 1, it must be calculated as the ensemble average of each component or divided into each frequency section and calculated in the form of a moving average.
앙상블 평균을 통해 산정된 코히어런스는 두 시간이력의 특정 주파수 성분에서 앙상블 샘플들 간의 위상차(phase difference)와 진폭의 비가 동일하게 유지될 때 1이 된다. 즉, 코히어런스는 앙상블들 간에서 위상차와 진폭의 비가 얼마나 잘 유지되는가를 나타내는 간접적인 지표에 불과하다.Coherence calculated through ensemble average becomes 1 when the phase difference and amplitude ratio between ensemble samples at specific frequency components of two time histories remain the same. In other words, coherence is only an indirect indicator of how well the ratio of phase difference and amplitude is maintained between ensembles.
특히, 등록특허 제10-2354932호에서는 두 방향의 풍하중들 간 상관성을 반영하고, 어느 하나의 풍하중이 최댓값일 때 다른 풍하중의 최댓값에 대한 비율을 조절함으로써, 풍동실험에 의존하지 않고 실제에 가까운 인공 시간이력하중을 생성할 수 있다. 그러나 상기 방법은 코히어런스로부터 a와 b라는 계수를 대략적으로 조정하여 상관성을 추정하므로 엔지니어의 개입이 필요하고, 원하는 수준의 주파수별 상관성에 도달하기 위해서는 많은 시행착오가 필요하다.In particular, Registered Patent No. 10-2354932 reflects the correlation between wind loads in two directions and adjusts the ratio to the maximum value of the other wind load when one wind load is the maximum value, thereby creating an artificial artificial intelligence that is close to reality without relying on wind tunnel experiments. Time history loads can be created. However, since the above method estimates correlation by roughly adjusting the coefficients a and b from coherence, an engineer's intervention is required, and a lot of trial and error is required to reach the desired level of correlation for each frequency.
상기와 같은 문제점을 해결하기 위하여 본 발명은 두 방향 풍하중에 대한 시간이력의 위상차를 직접 이용하여 원하는 수준의 상관성을 정확하게 재생함으로써, 풍동실험에 의존하지 않고 실제에 가까운 인공 시간이력하중을 생성할 수 있는 상관성을 고려한 시간이력풍하중 산정 방법을 제공하고자 한다.In order to solve the above problems, the present invention directly uses the phase difference of the time history for two-way wind load to accurately reproduce the desired level of correlation, making it possible to generate an artificial time history load that is close to reality without relying on wind tunnel experiments. The purpose of this study is to provide a time history wind load calculation method that takes into account the existing correlation.
바람직한 실시예에 따른 본 발명은 풍방향 하중, 풍직각 방향 하중 및 비틀림 방향 하중 중 어느 하나인 제1풍하중과, 나머지 하중 중 어느 하나인 제2풍하중에 대해 시간이력풍하중을 재생하기 위해 컴퓨팅 장치에 의해 수행되는 것으로, (a) 제1풍하중에 대한 임의의 제1랜덤 위상각과 제2풍하중에 대한 제2랜덤 위상각의 위상차에 대한 앙상블 평균(
)을 가정하는 단계; (b) 상기 앙상블 평균(
)에 의해 앙상블 표준편차(
)를 산정하는 단계; (c) 상기 위상차의 앙상블 표준편차(
)에 의해 위상차에 대한 앙상블 평균(
)을 보정하는 단계; (d) 상기 보정된 위상차의 앙상블 평균과 상기 위상차의 앙상블 표준편차(
)에 의해 각 주파수에서 제1시간이력풍하중과 제2시간이력풍하중의 위상차(
)를 산정하는 단계; (e) 제1시간이력에 대한 균등분포를 갖는 랜덤 제1위상각 세트(
)를 재생하는 단계; (f) 상기 (d) 단계에서 산정된 위상차(
)와 상기 (e) 단계에서 생성된 랜덤 제1위상각 세트(
)로부터 제2시간이력에 대한 제2위상각 세트(
)를 산정하는 단계; 및 (g) 상기 제1위상각 세트(
)와 제2위상각 세트(
)에 의해 각각 제1시간이력풍하중(X(t))과 제2시간이력풍하중(Y(t))을 재생하는 단계; 로 구성되는 것을 특징으로 하는 상관성을 고려한 시간이력풍하중 산정 방법을 제공한다. The present invention according to a preferred embodiment is provided to a computing device to reproduce the time history wind load for the first wind load, which is one of the wind direction load, the wind angle load, and the torsional load, and the second wind load, which is one of the remaining loads. This is performed by (a) the ensemble average of the phase difference between a random first random phase angle for the first wind load and a second random phase angle for the second wind load ( ) assuming that; (b) the ensemble average ( ) by the ensemble standard deviation ( ) calculating; (c) ensemble standard deviation of the phase difference ( ) by the ensemble average for the phase difference ( ) correcting; (d) the ensemble mean of the corrected phase difference and the ensemble standard deviation of the phase difference ( ), the phase difference between the first time history wind load and the second time history wind load at each frequency ( ) calculating; (e) Random first phase angle set with uniform distribution for the first time history ( ) to reproduce; (f) phase difference calculated in step (d) above ( ) and the random first phase angle set generated in step (e) ( ) from the second phase angle set for the second time history ( ) calculating; and (g) the first phase angle set ( ) and the second phase angle set ( ) Reproducing the first time history wind load (X(t)) and the second time history wind load (Y(t)), respectively; It provides a time history wind load calculation method considering correlation, which is characterized by consisting of.
다른 바람직한 실시예에 따른 본 발명은 상기 (a) 단계는 풍동실험 데이터로부터 위상차의 코사인 값에 대한 앙상블 평균(
) 데이터를 제공받아 상기 위상차의 코사인 값에 대한 앙상블 평균에 의해 위상차의 앙상블 평균(
)을 아래 [수학식] 에 의해 가정하는 것을 특징으로 하는 상관성을 고려한 시간이력풍하중 산정 방법을 제공한다. In the present invention according to another preferred embodiment, step (a) is the ensemble average of the cosine value of the phase difference from the wind tunnel experiment data ( ) Data is provided and the ensemble average of the phase difference is calculated by the ensemble average of the cosine value of the phase difference ( ) provides a time history wind load calculation method considering correlation, which is characterized by assuming the equation below.
[수학식][Equation]
다른 바람직한 실시예에 따른 본 발명은 상기 (c) 단계는 아래 [수학식] 에 의해 앙상블 평균(
)을 보정하는 것을 특징으로 하는 상관성을 고려한 시간이력풍하중 산정 방법을 제공한다. In the present invention according to another preferred embodiment, the step (c) is performed by the ensemble average ( ) provides a time history wind load calculation method considering correlation, which is characterized by correcting.
[수학식][Equation]
여기서 좌변의 
는 보정된 앙상블 평균, 보정계수 
, 
.Here on the left side is the corrected ensemble mean, correction coefficient , .
다른 바람직한 실시예에 따른 본 발명은 상기 (b) 단계는 아래 [수학식] 에 의해 앙상블 표준편차(
)를 산정하는 것을 특징으로 하는 상관성을 고려한 시간이력풍하중 산정 방법을 제공한다. In the present invention according to another preferred embodiment, step (b) is performed by calculating the ensemble standard deviation ( ) provides a time history wind load calculation method considering correlation, which is characterized by calculating ).
[수학식][Equation]
여기서 
, 
, 
는 위상차의 코사인의 앙상블 표준편차.here , , is the ensemble standard deviation of the cosine of the phase difference.
본 발명에 따르면 두 방향 풍하중에 대한 시간이력의 위상차를 직접 이용하여 원하는 수준의 상관성을 정확하게 재생함으로써 풍동실험에 의존하지 않고 실제에 가까운 인공 시간이력하중을 생성할 수 있는 상관성을 고려한 시간이력풍하중 산정 방법을 제공할 수 있다. According to the present invention, time history wind load calculation considering correlation can generate an artificial time history load close to reality without relying on wind tunnel experiments by accurately reproducing the desired level of correlation by directly using the phase difference of the time history for two-way wind load. A method can be provided.
이에 따라 풍동실험에 매번 의존하지 않고도 많은 연구자에 의해 정립된 파워스펙트럼밀도 함수와 위상차 정보를 활용하여 정확한 시간이력하중을 생성할 수 있다.Accordingly, accurate time history loads can be generated by utilizing the power spectrum density function and phase difference information established by many researchers without having to rely on wind tunnel experiments every time.
또한, 현재 건축구조기준에서 사용 중인 등가정적하중과 달리 정확한 시간이력해석에 의해 구조물의 응답을 산정함으로써 구조물의 진동 모드를 보다 정확하게 반영할 수 있으며, 등가정적하중을 사용할 수 없는 비선형 해석을 수행할 수 있다.In addition, unlike the equivalent static load currently used in building structural standards, the vibration mode of the structure can be more accurately reflected by calculating the response of the structure through accurate time history analysis, and nonlinear analysis in which the equivalent static load cannot be performed can be performed. You can.
도 1은 구조물에 작용하는 풍하중의 종류를 도시하는 도면.1 is a diagram showing types of wind load acting on a structure.
도 2는 CPSD 함수의 실수부와 허수부의 관계에 의해 정의되는 위상차를 나타내는 그래프.Figure 2 is a graph showing the phase difference defined by the relationship between the real part and the imaginary part of the CPSD function.
도 3은 위상차와 위상차의 코사인의 앙상블 평균의 관계를 나타내는 그래프.Figure 3 is a graph showing the relationship between the ensemble average of the phase difference and the cosine of the phase difference.
도 4는 위상차와 위상차의 코사인의 앙상블 표준편차의 관계를 나타내는 그래프.Figure 4 is a graph showing the relationship between the phase difference and the ensemble standard deviation of the cosine of the phase difference.
도 5는 위상차 정보로부터 상관성이 있는 시간이력을 재생하는 절차를 나타내는 순서도.5 is a flowchart showing a procedure for reproducing a correlated time history from phase difference information.
도 6은 재생된 위상차 코사인의 평균의 검증 결과를 나타내는 그래프.Figure 6 is a graph showing the verification results of the average of the reproduced phase difference cosine.
도 7은 재생된 위상차 코사인의 표준편차의 검증 결과를 나타내는 그래프.Figure 7 is a graph showing the verification results of the standard deviation of the reproduced phase difference cosine.
도 8은 풍동실험 결과와 재생된 위상차 샘플들의 비교 결과를 나타내는 그래프.Figure 8 is a graph showing the comparison results of wind tunnel experiment results and reproduced phase difference samples.
상기와 같은 목적을 달성하기 위하여 본 발명의 상관성을 고려한 시간이력풍하중 산정 방법은 풍방향 하중, 풍직각 방향 하중 및 비틀림 방향 하중 중 어느 하나인 제1풍하중과, 나머지 하중 중 어느 하나인 제2풍하중에 대해 시간이력풍하중을 재생하기 위해 컴퓨팅 장치에 의해 수행되는 것으로, (a) 제1풍하중에 대한 임의의 제1랜덤 위상각과 제2풍하중에 대한 제2랜덤 위상각의 위상차에 대한 앙상블 평균(
)을 가정하는 단계; (b) 상기 앙상블 평균(
)에 의해 앙상블 표준편차(
)를 산정하는 단계; (c) 상기 위상차의 앙상블 표준편차(
)에 의해 위상차에 대한 앙상블 평균(
)을 보정하는 단계; (d) 상기 보정된 위상차의 앙상블 평균과 상기 위상차의 앙상블 표준편차(
)에 의해 각 주파수에서 제1시간이력풍하중과 제2시간이력풍하중의 위상차(
)를 산정하는 단계; (e) 제1시간이력에 대한 균등분포를 갖는 랜덤 제1위상각 세트(
)를 재생하는 단계; (f) 상기 (d) 단계에서 산정된 위상차(
)와 상기 (e) 단계에서 생성된 랜덤 제1위상각 세트(
)로부터 제2시간이력에 대한 제2위상각 세트(
)를 산정하는 단계; 및 (g) 상기 제1위상각 세트(
)와 제2위상각 세트(
)에 의해 각각 제1시간이력풍하중(X(t))과 제2시간이력풍하중(Y(t))을 재생하는 단계; 로 구성되는 것을 특징으로 한다.In order to achieve the above object, the time history wind load calculation method considering the correlation of the present invention is the first wind load, which is one of the wind direction load, wind angle direction load, and torsional load, and the second wind load, which is one of the remaining loads. is performed by a computing device to reproduce the time history wind load for (a) the ensemble average of the phase difference between a random first random phase angle for the first wind load and a second random phase angle for the second wind load ( ) assuming that; (b) the ensemble average ( ) by the ensemble standard deviation ( ) calculating; (c) ensemble standard deviation of the phase difference ( ) by the ensemble average for the phase difference ( ) correcting; (d) the ensemble mean of the corrected phase difference and the ensemble standard deviation of the phase difference ( ), the phase difference between the first time history wind load and the second time history wind load at each frequency ( ) calculating; (e) Random first phase angle set with uniform distribution for the first time history ( ) to reproduce; (f) phase difference calculated in step (d) above ( ) and the random first phase angle set generated in step (e) ( ) from the second phase angle set for the second time history ( ) calculating; and (g) the first phase angle set ( ) and the second phase angle set ( ) Reproducing the first time history wind load (X(t)) and the second time history wind load (Y(t)), respectively; It is characterized by being composed of.
이하, 첨부한 도면 및 바람직한 실시예에 따라 본 발명을 상세히 설명한다.Hereinafter, the present invention will be described in detail according to the accompanying drawings and preferred embodiments.
도 2는 CPSD 함수의 실수부와 허수부의 관계에 의해 정의되는 위상차를 나타내는 그래프이다. Figure 2 is a graph showing the phase difference defined by the relationship between the real part and the imaginary part of the CPSD function.
본 발명 상관성을 고려한 시간이력풍하중 산정 방법은 풍방향 하중, 풍직각 방향 하중 및 비틀림 방향 하중 중 어느 하나인 제1풍하중과, 나머지 하중 중 어느 하나인 제2풍하중에 대해 시간이력풍하중을 재생하기 위해 컴퓨팅 장치에 의해 수행되는 것으로, (a) 제1풍하중에 대한 임의의 제1랜덤 위상각과 제2풍하중에 대한 제2랜덤 위상각의 위상차에 대한 앙상블 평균(
)을 가정하는 단계; (b) 상기 앙상블 평균(
)에 의해 앙상블 표준편차(
)를 산정하는 단계; (c) 상기 위상차의 앙상블 표준편차(
)에 의해 위상차에 대한 앙상블 평균(
)을 보정하는 단계; (d) 상기 보정된 위상차의 앙상블 평균과 상기 위상차의 앙상블 표준편차(
)에 의해 각 주파수에서 제1시간이력풍하중과 제2시간이력풍하중의 위상차(
)를 산정하는 단계; (e) 제1시간이력에 대한 균등분포를 갖는 랜덤 제1위상각 세트(
)를 재생하는 단계; (f) 상기 (d) 단계에서 산정된 위상차(
)와 상기 (e) 단계에서 생성된 랜덤 제1위상각 세트(
)로부터 제2시간이력에 대한 제2위상각 세트(
)를 산정하는 단계; 및 (g) 상기 제1위상각 세트(
)와 제2위상각 세트(
)에 의해 각각 제1시간이력풍하중(X(t))과 제2시간이력풍하중(Y(t))을 재생하는 단계; 로 구성되는 것을 특징으로 한다.The time history wind load calculation method considering the correlation of the present invention is to reproduce the time history wind load for the first wind load, which is one of the wind direction load, wind angle direction load, and torsional load, and the second wind load, which is one of the remaining loads. Performed by a computing device, (a) an ensemble average of the phase difference between a random first random phase angle for the first wind load and a second random phase angle for the second wind load ( ) assuming that; (b) the ensemble average ( ) by the ensemble standard deviation ( ) calculating; (c) ensemble standard deviation of the phase difference ( ) by the ensemble average for the phase difference ( ) correcting; (d) the ensemble mean of the corrected phase difference and the ensemble standard deviation of the phase difference ( ), the phase difference between the first time history wind load and the second time history wind load at each frequency ( ) calculating; (e) Random first phase angle set with uniform distribution for the first time history ( ) to reproduce; (f) phase difference calculated in step (d) above ( ) and the random first phase angle set generated in step (e) ( ) from the second phase angle set for the second time history ( ) calculating; and (g) the first phase angle set ( ) and the second phase angle set ( ) Reproducing the first time history wind load (X(t)) and the second time history wind load (Y(t)), respectively; It is characterized by being composed of.
본 발명은 두 방향 풍하중에 대한 시간이력의 위상차를 직접 이용하여 원하는 수준의 상관성을 정확하게 재생함으로써, 풍동실험에 의존하지 않고 실제에 가까운 인공 시간이력하중을 생성할 수 있는 상관성을 고려한 시간이력풍하중 산정 방법을 제공하기 위한 것이다. The present invention accurately reproduces the desired level of correlation by directly using the phase difference of the time history for the two-way wind load, and calculates the time history wind load considering the correlation that can generate an artificial time history load close to reality without relying on wind tunnel experiments. It is intended to provide a method.
본 발명은 풍방향 하중, 풍직각 방향 하중 및 비틀림 방향 하중 중 어느 하나인 제1풍하중과, 나머지 하중 중 어느 하나인 제2풍하중에 대해 시간이력풍하중을 재생하기 위해 컴퓨팅 장치에 의해 수행된다. The present invention is performed by a computing device to reproduce a time history wind load for a first wind load, which is one of a wind direction load, a wind angle load, and a twisting direction load, and a second wind load, which is one of the remaining loads.
두 시간이력의 위상차는 풍동실험데이터의 크로스파워스펙트럼밀도(Cross Power Spectral Density, CPSD) 함수로부터 추출될 수 있다. 일반적으로 CPSD 함수는 아래 [수학식 3] 과 같이 복소수이다.The phase difference between the two time histories can be extracted from the Cross Power Spectral Density (CPSD) function of the wind tunnel experiment data. In general, the CPSD function is a complex number as shown in [Equation 3] below.
[수학식 3][Equation 3]
여기서, 
는 CPSD 함수의 실수부(Real part), 
는 CPSD 함수의 허수부(Imaginary part) 이다.here, is the real part of the CPSD function, is the imaginary part of the CPSD function.
도 2 및 아래 [수학식 4] 와 같이, 위상차는 CPSD 함수의 실수부와 허수부의 관계에 의해 정의될 수 있으며, [수학식 4] 에 의해 위상차는 [수학식 5] 와 같이 나타낼 수 있다.As shown in FIG. 2 and [Equation 4] below, the phase difference can be defined by the relationship between the real part and the imaginary part of the CPSD function, and by [Equation 4], the phase difference can be expressed as [Equation 5].
[수학식 4][Equation 4]
[수학식 5][Equation 5]
위상차 산정 시 역탄젠트 함수를 사용할 때는 실수부와 허수부의 부호를 고려하여 4분면 역탄젠트를 사용한다.When using the inverse tangent function when calculating the phase difference, use the four-quadrant inverse tangent by considering the signs of the real and imaginary parts.
이렇게 산정된 위상차 정보는 -π에서 π 사이의 값을 갖게 된다.The phase difference information calculated in this way has a value between -π and π.
시간 영역에서 랜덤 프로세스에 가까운 풍하중의 특성은 위상차 정보에서도 비슷하게 나타나 경향성을 파악하기 어렵다.The characteristics of wind load, which is close to a random process in the time domain, appear similarly in phase difference information, making it difficult to identify trends.
따라서 풍하중의 위상차 정보에 대한 경향성을 파악하기 위해 앙상블 평균(
) 및 앙상블 표준편차(
)를 이용한다. Therefore, to determine the tendency of the phase difference information of wind load, the ensemble average ( ) and ensemble standard deviation ( ) is used.
그러나 앙상블 평균(
)만으로는 경향성을 제대로 파악하기 어려울 뿐 아니라 앙상블 평균(
)과 앙상블 표준편차(
)를 위상차 정보로 변환하는 과정에서 평균과 표준편차가 서로 영향을 끼친다.However, the ensemble average ( ) alone is not only difficult to properly identify trends, but also the ensemble average ( ) and ensemble standard deviation ( ) in the process of converting phase difference information, the average and standard deviation affect each other.
따라서 상기 (a) 단계에서는 위상차에 대한 앙상블 평균(
)을 초기 설정 값으로 먼저 가정한다.Therefore, in step (a), the ensemble average for the phase difference ( ) is first assumed as the initial setting value.
즉, 상기 (a) 단계에서는 제1풍하중에 대한 임의의 제1랜덤 위상각과 제2풍하중에 대한 제2랜덤 위상각의 위상차에 대한 앙상블 평균(
)을 가정한다.That is, in step (a), the ensemble average of the phase difference between the random first random phase angle for the first wind load and the second random phase angle for the second wind load ( ) is assumed.
이후, (b) 상기 앙상블 평균(
)에 의해 앙상블 표준편차(
)를 선정하고, (c) 상기 산정된 위상차의 앙상블 표준편차(
)에 의해 위상차에 대한 앙상블 평균(
)을 보정한다. Afterwards, (b) the ensemble average ( ) by the ensemble standard deviation ( ) is selected, and (c) the ensemble standard deviation of the calculated phase difference ( ) by the ensemble average for the phase difference ( ) is corrected.
상기 (b) 단계와 (c) 단계는 필요에 따라 수회 반복 수행할 수도 있다.Steps (b) and (c) may be repeated as many times as necessary.
다음으로, (d) 상기 (c) 단계에서 보정된 위상차의 앙상블 평균 및 위상차의 앙상블 표준편차(
)에 의해 각 주파수에서 두 시간이력풍하중인 제1시간이력풍하중과 제2시간이력풍하중의 위상차(
)를 산정한다.Next, (d) the ensemble average of the phase difference corrected in step (c) and the ensemble standard deviation of the phase difference ( ), the phase difference between the first time history wind load and the second time history wind load, which are two time history wind loads at each frequency ( ) is calculated.
이때, 앙상블 평균과 앙상블 표준편차(
)에 의해 정규분포를 따르는 난수를 생성함으로써 위상차 세트(
)를 생성한다. At this time, the ensemble mean and ensemble standard deviation ( By generating random numbers that follow a normal distribution by ), the phase difference set ( ) is created.
그리고 (e) 제1시간이력에 대한 균등분포를 갖는 랜덤 제1위상각 세트(
)를 재생하고, (f) 상기 (d) 단계에서 산정된 위상차(
)와 상기 (e) 단계에서 생성된 랜덤 제1위상각 세트(
)로부터 제2시간이력에 대한 제2위상각 세트(
)를 산정한다. And (e) a random first phase angle set with a uniform distribution for the first time history ( ) and (f) the phase difference calculated in step (d) ( ) and the random first phase angle set generated in step (e) ( ) from the second phase angle set for the second time history ( ) is calculated.
상기 (e) 단계에서는 기준이 되는 시간이력풍하중인 제1시간이력풍하중(X(t))에 대해 균등분포를 갖고 있는 랜덤 제1위상각 세트(
)를 재생한다. In step (e), a random first phase angle set ( ) play.
상기 (f) 단계에서는 상기 (e) 단계에서 생성된 제1위상각 세트(
)와 상기 (d) 단계에서 생성된 위상차 세트에 의해 제2위상각 세트(
)를 산정한다.In step (f), the first phase angle set generated in step (e) ( ) and a second phase angle set by the phase difference set generated in step (d) ( ) is calculated.
마지막으로 (g) 상기 제1위상각 세트(
)와 제2위상각 세트(
)에 의해 각각 제1시간이력풍하중(X(t))과 제2시간이력풍하중(Y(t))을 재생한다. Finally, (g) the first phase angle set ( ) and the second phase angle set ( ) to reproduce the first time history wind load (X(t)) and the second time history wind load (Y(t)), respectively.
상기 제1시간이력풍하중(X(t)) 및 제2시간이력풍하중(Y(t))은 각각 아래 [수학식 6] 및 [수학식 7] 과 같다.The first time history wind load (X(t)) and the second time history wind load (Y(t)) are respectively shown in [Equation 6] and [Equation 7] below.
[수학식 6][Equation 6]
[수학식 7][Equation 7]
위와 같은 방법을 통해 풍동실험에 매번 의존하지 않고도 많은 연구자에 의해 정립된 파워스펙트럼밀도 함수와 위상차 정보를 활용하여 정확한 시간이력하중을 생성할 수 있다.Through the above method, an accurate time history load can be generated by utilizing the power spectrum density function and phase difference information established by many researchers without always relying on wind tunnel experiments.
이에 따라 현재 건축구조기준에서 사용 중인 등가정적하중과 달리 정확한 시간이력해석에 의해 구조물의 응답을 산정함으로써, 구조물의 진동 모드를 보다 정확하게 반영할 수 있다. 아울러 등가정적하중을 사용할 수 없는 비선형 해석을 수행할 수 있다. Accordingly, unlike the equivalent static load currently used in building structural standards, the vibration mode of the structure can be reflected more accurately by calculating the response of the structure through accurate time history analysis. In addition, nonlinear analysis can be performed where equivalent static loads cannot be used.
상기 (a) 단계는 풍동실험 데이터로부터 위상차의 코사인 값에 대한 앙상블 평균(
) 데이터를 제공받아 상기 위상차의 코사인 값에 대한 앙상블 평균에 의해 위상차의 앙상블 평균(
)을 아래 [수학식] 에 의해 가정할 수 있다.The step (a) is the ensemble average of the cosine value of the phase difference from the wind tunnel experiment data ( ) Data is provided and the ensemble average of the phase difference is calculated by the ensemble average of the cosine value of the phase difference ( ) can be assumed by the [mathematical formula] below.
[수학식][Equation]
위상차 정보에 대해 직접적으로 앙상블 평균(
)이나 앙상블 표준편차(
)를 산정할 경우, 위상차의 주기적 특성 때문에 그 결과가 왜곡될 수 있다.Ensemble average ( ) or ensemble standard deviation ( ), the results may be distorted due to the periodic nature of the phase difference.
예를 들어, 주파수 영역에서의 상관성으로 인해 위상차 평균이 π에 가까운 경우, 약간의 변동으로 이를 초과할 경우 -π에 가까운 값을 나타내게 된다. 이로 인해 앙상블 평균(
)과 앙상블 표준편차(
)가 왜곡되는 것이다. For example, when the average phase difference is close to π due to correlation in the frequency domain, if it exceeds this with slight fluctuation, it shows a value close to -π. This results in the ensemble average ( ) and ensemble standard deviation ( ) is distorted.
따라서 이를 해결하기 위해 위상차 정보를 코사인 함수의 형태로 제시할 수 있다. 코사인 함수를 적용하면 위와 같은 주기적 성질에 의한 왜곡을 방지할 수 있다.Therefore, to solve this problem, phase difference information can be presented in the form of a cosine function. By applying the cosine function, distortion due to the above periodic properties can be prevented.
즉, 풍동실험 데이터로부터 확보된 위상차 데이터를 그대로 사용하는 것이 아니라 위상차의 코사인 값에 대한 앙상블 평균 데이터를 활용한다.In other words, rather than using the phase difference data obtained from wind tunnel experiment data as is, the ensemble average data for the cosine value of the phase difference is utilized.
풍동실험 데이터에서 제공된 위상차의 앙상블 평균(
)에 대한 코사인 값에 대해 역코사인 함수를 적용하여 초기 앙상블 평균(
)으로 가정한다.Ensemble average of phase differences provided by wind tunnel test data ( ) by applying the inverse cosine function to the cosine value for the initial ensemble mean ( ) is assumed.
이에 따라 상기 (a) 단계에서 앙상블 평균(
) 가정을 위한 초깃값은 아래 [수학식 8] 과 같이 나타낼 수 있다.Accordingly, in step (a) above, the ensemble average ( ) The initial value for the assumption can be expressed as [Equation 8] below.
[수학식 8][Equation 8]
이때, 풍동실험을 통해 얻은 위상차를 위상차의 코사인으로 변환하는 과정에서 그 부호 정보가 사라지게 된다. 그렇기 때문에 이는 실험결과로부터 추정하거나, 두 시간이력의 최댓값 발생의 순서가 상관없는 경우 이를 무시하고 산정할 수 있다.At this time, in the process of converting the phase difference obtained through the wind tunnel experiment into the cosine of the phase difference, the sign information disappears. Therefore, this can be estimated from the experimental results, or ignored if the order of occurrence of the maximum value of the two time histories is not related.
다만, 세 개 이상의 시간이력에 대해 상관성을 재생할 경우에는 서로의 위상차의 코사인의 앙상블 평균값이 충족되도록 부호를 정한다.However, when reproducing the correlation for three or more time histories, the sign is set so that the ensemble average value of the cosine of each other's phase differences is satisfied.
도 3은 위상차와 위상차의 코사인의 앙상블 평균의 관계를 나타내는 그래프이다.Figure 3 is a graph showing the relationship between the phase difference and the ensemble average of the cosine of the phase difference.
상기 (c) 단계는 아래 [수학식] 에 의해 앙상블 평균(
)을 보정할 수 있다. Step (c) above is performed by the ensemble average ( ) can be corrected.
[수학식][Equation]
여기서, 좌변의 
는 보정된 앙상블 평균, 보정계수 
, 
.Here, on the left side is the corrected ensemble mean, correction coefficient , .
위상차의 앙상블 평균(
)과 위상차의 코사인의 앙상블 평균(
)은 도 3과 같은 관계가 있다.Ensemble average of phase difference ( ) and the ensemble mean of the cosine of the phase difference ( ) has the same relationship as in Figure 3.
따라서 위상차의 코사인의 앙상블 평균(
)으로부터 아래 [수학식 9] 를 통해 위상차의 앙상블 평균(
)을 추정할 수 있다.Therefore, the ensemble mean of the cosine of the phase difference ( ) from the ensemble average of the phase difference through [Equation 9] below ( ) can be estimated.
[수학식 9][Equation 9]
여기서, 좌변의 
는 보정된 앙상블 평균이고, 보정계수 
, 
이다.Here, on the left side is the corrected ensemble mean, and the correction coefficient , am.
이때, 역코사인의 값이 실수(real number)로 정의되기 위해서는 
가 -1에서 1 사이의 값을 가져야 하며, 이를 초과할 경우 그 값을 -1 또는 1로 제한해야 한다.At this time, in order for the value of the inverse cosine to be defined as a real number, must have a value between -1 and 1, and if it exceeds this value, the value must be limited to -1 or 1.
전술한 바와 같이, 위상차의 코사인 정보(앙상블 평균(
)과 앙상블 표준편차(
))를 위상차의 정보로 변환하는 과정에서 평균과 표준편차는 서로 영향을 준다. 따라서 초기 설정 값으로 위상차의 평균을 [수학식 8] 과 같이 산정한 후 0회 또는 1회 정도의 최소한의 반복 과정을 통해 보정할 수도 있다.As mentioned above, the cosine information of the phase difference (ensemble average ( ) and ensemble standard deviation ( In the process of converting )) into phase difference information, the average and standard deviation affect each other. Therefore, the average phase difference can be calculated as the initial setting value as in [Equation 8] and then corrected through a minimum repetition process of 0 or 1 time.
도 4는 위상차와 위상차의 코사인의 앙상블 표준편차의 관계를 나타내는 그래프이다.Figure 4 is a graph showing the relationship between the phase difference and the ensemble standard deviation of the cosine of the phase difference.
상기 (b) 단계는 아래 [수학식] 에 의해 앙상블 표준편차(
)를 산정할 수 있다. In step (b), the ensemble standard deviation ( ) can be calculated.
[수학식][Equation]
여기서 
, 
, 
는 위상차의 코사인의 앙상블 표준편차.here , , is the ensemble standard deviation of the cosine of the phase difference.
위상차 정보를 코사인 함수로 표현하더라도 앙상블 평균(
)만으로는 그 경향성을 제대로 나타낼 수 없다. 서로 상관성이 없는 경우 위상차는 -π에서 π 사이에서 등분포 형태로 나타나며, 위상차의 코사인의 앙상블 평균은 0이 된다.Even if the phase difference information is expressed as a cosine function, the ensemble average ( ) alone cannot properly represent the tendency. If there is no correlation, the phase difference appears uniformly distributed between -π and π, and the ensemble average of the cosine of the phase difference is 0.
이 경우 위상차의 코사인의 앙상블 평균으로부터 역코사인을 그대로 적용하면 ±π/2의 값이 나타난다.In this case, if the inverse cosine is applied as is from the ensemble average of the cosine of the phase difference, a value of ±π/2 appears.
따라서 이러한 경우를 위해 앙상블 표준편차(
) 정보가 같이 제시되어야 한다.Therefore, for these cases, the ensemble standard deviation ( ) information must be presented together.
위상차가 -π에서 π 사이에서 등분포 형태일 경우, 위상차의 코사인은 아래 [수학식 10] 과 같은 확률밀도함수(PDF)를 갖는다.When the phase difference is uniformly distributed between -π and π, the cosine of the phase difference has a probability density function (PDF) as shown in [Equation 10] below.
[수학식 10][Equation 10]
이때, 위상차의 코사인의 표준편차는 
가 된다.At this time, the standard deviation of the cosine of the phase difference is It becomes.
반면, 위상차 간에 상관성이 있을 경우, 위상차 앙상블 사이에서 특정한 평균값과 표준편차를 갖는 Gaussian 분포를 가질 것이라고 가정할 수 있다. 하지만 위상차의 코사인의 앙상블 평균과 앙상블 표준편차 형태로 풍동실험 결과가 주어졌기 때문에, 이들로부터 위상차 정보를 복원하기 위해서는 이들 간의 관계를 정의해야 한다.On the other hand, if there is correlation between phase differences, it can be assumed that the phase difference ensemble will have a Gaussian distribution with a specific mean value and standard deviation. However, since the results of the wind tunnel experiment are given in the form of the ensemble mean and ensemble standard deviation of the cosine of the phase difference, the relationship between them must be defined in order to restore the phase difference information from them.
Gaussian 분포를 갖는 경우에 대해 위상차의 코사인의 분포를 이론적으로는 유도하는 것은 매우 어렵기 때문에, 본 발명에서는 10,000개의 위상차를 만들어서 그 관계를 정의하는 약산식을 도입하였다.Since it is very difficult to theoretically derive the distribution of the cosine of the phase difference in the case of Gaussian distribution, the present invention creates 10,000 phase differences and introduces an abbreviation equation that defines the relationship.
우선 위상차와 위상차의 코사인의 앙상블 표준편차들(
, 
)의 관계는 도 4와 같다.First, the ensemble standard deviations of the phase difference and the cosine of the phase difference ( , ) The relationship is as shown in Figure 4.
전술한 바와 같이, 서로 상관성이 없어질 경우(위상차의 표준편차가 증가할 경우), 등분포 위상차의 코사인의 앙상블 표준편차의 이론값인 
에 수렴해 가는 것을 도 4에서 확인할 수 있다. 이 둘의 관계는 위상차의 평균에 의해 영향을 받기 때문에, 위상차의 평균(
)으로부터 이를 약산하기 위해 이선형(bilinear) 곡선의 꺾인점(
, 
)을 아래 [수학식 11] 과 같이 계산할 수 있다.As described above, when correlation disappears (when the standard deviation of the phase difference increases), the theoretical value of the ensemble standard deviation of the cosine of the uniformly distributed phase difference is It can be seen in Figure 4 that it is converging to . Since the relationship between the two is influenced by the average of the phase difference, the average of the phase difference ( ), the bending point of the bilinear curve ( , ) can be calculated as in [Equation 11] below.
[수학식 11][Equation 11]
이에 따라 위상차와 위상차의 코사인의 앙상블 표준편차(
) 관계로부터 위상차의 앙상블 표준편차(
)는 아래 [수학식 12] 와 같이 추정될 수 있다.Accordingly, the ensemble standard deviation of the phase difference and the cosine of the phase difference ( ) ensemble standard deviation of the phase difference from the relationship ( ) can be estimated as shown in [Equation 12] below.
[수학식 12][Equation 12]
도 5는 위상차 정보로부터 상관성이 있는 시간이력을 재생하는 절차를 나타내는 순서도이다.Figure 5 is a flowchart showing a procedure for reproducing a correlated time history from phase difference information.
도 5에는 본 발명 상관성을 고려한 시간이력풍하중 산정 방법이 순서도로 도시된다. 도 5와 같은 순서를 통하여 위상차 정보로부터 시간이력풍하중 X(t), Y(t)를 구할 수 있다. Figure 5 shows a flowchart showing a time history wind load calculation method considering the correlation of the present invention. Time history wind loads X(t) and Y(t) can be obtained from phase difference information through the procedure shown in FIG. 5.
도 6 내지 도 8은 위상차 정보의 복원 검증 결과를 나타내는 그래프들이다.Figures 6 to 8 are graphs showing the results of restoration and verification of phase difference information.
구체적으로 도 6은 재생된 위상차 코사인의 평균의 검증 결과를 나타내고, 도 7은 재생된 위상차 코사인의 표준편차의 검증 결과를 나타낸다. 여기에서 N은 반복횟수를 나타낸다. Specifically, Figure 6 shows the verification result of the average of the reproduced phase difference cosine, and Figure 7 shows the verification result of the standard deviation of the reproduced phase difference cosine. Here, N represents the number of repetitions.
그리고 도 8은 풍동실험 결과와 재생된 위상차 샘플들의 비교 결과를 나타낸다. And Figure 8 shows the comparison results of the wind tunnel test results and the reproduced phase difference samples.
도 6 내지 도 8을 통하여 재생 과정에서 Gaussian 분포를 활용하여 재생하기 때문에 풍동실험 결과 본래의 위상차 범위인 -π~π를 벗어난 결과가 발생하긴 하지만, 위상차의 주기적 특성으로 인해 2π 단위의 사이클을 보정한 것과 결과적으로 동일하게 되는 것을 알 수 있다.6 to 8, since the Gaussian distribution is used in the regeneration process to reproduce, the wind tunnel experiment results in a result outside the original phase difference range of -π to π occur. However, due to the periodic nature of the phase difference, the cycle in units of 2π is corrected. You can see that the result is the same as what was done.
본 발명의 상관성을 고려한 시간이력풍하중 산정 방법은 두 방향 풍하중에 대한 시간이력의 위상차를 직접 이용하여 원하는 수준의 상관성을 정확하게 재생함으로써 풍동실험에 의존하지 않고 실제에 가까운 인공 시간이력하중을 생성할 수 있다는 점에서 산업상 이용 가능성이 있다.The time history wind load calculation method considering the correlation of the present invention can generate an artificial time history load close to reality without relying on wind tunnel experiments by accurately reproducing the desired level of correlation by directly using the phase difference of the time history for two-way wind load. There is potential for industrial use in that it exists.
Claims (4)
- 풍방향 하중, 풍직각 방향 하중 및 비틀림 방향 하중 중 어느 하나인 제1풍하중과, 나머지 하중 중 어느 하나인 제2풍하중에 대해 시간이력풍하중을 재생하기 위해 컴퓨팅 장치에 의해 수행되는 것으로, It is performed by a computing device to reproduce the time history wind load for the first wind load, which is one of the wind direction load, the wind angle load, and the torsional load, and the second wind load, which is one of the remaining loads,(a) 제1풍하중에 대한 임의의 제1랜덤 위상각과 제2풍하중에 대한 제2랜덤 위상각의 위상차에 대한 앙상블 평균()을 가정하는 단계; (a) Ensemble average of the phase difference between a random first random phase angle for the first wind load and a second random phase angle for the second wind load (
) assuming that;(b) 상기 앙상블 평균()에 의해 앙상블 표준편차(
)를 산정하는 단계;(b) the ensemble average (
) by the ensemble standard deviation () calculating;(c) 상기 위상차의 앙상블 표준편차()에 의해 위상차에 대한 앙상블 평균(
)을 보정하는 단계;(c) ensemble standard deviation of the phase difference (
) by the ensemble average for the phase difference () correcting;(d) 상기 보정된 위상차의 앙상블 평균과 상기 위상차의 앙상블 표준편차()에 의해 각 주파수에서 제1시간이력풍하중과 제2시간이력풍하중의 위상차(
)를 산정하는 단계; (d) the ensemble mean of the corrected phase difference and the ensemble standard deviation of the phase difference (
), the phase difference between the first time history wind load and the second time history wind load at each frequency () calculating;(e) 제1시간이력에 대한 균등분포를 갖는 랜덤 제1위상각 세트()를 재생하는 단계;(e) Random first phase angle set with uniform distribution for the first time history (
) to reproduce;(f) 상기 (d) 단계에서 산정된 위상차()와 상기 (e) 단계에서 생성된 랜덤 제1위상각 세트(
)로부터 제2시간이력에 대한 제2위상각 세트(
)를 산정하는 단계; 및(f) phase difference calculated in step (d) above (
) and the random first phase angle set generated in step (e) () from the second phase angle set for the second time history () calculating; and(g) 상기 제1위상각 세트()와 제2위상각 세트(
)에 의해 각각 제1시간이력풍하중(X(t))과 제2시간이력풍하중(Y(t))을 재생하는 단계; 로 구성되는 것을 특징으로 하는 상관성을 고려한 시간이력풍하중 산정 방법.(g) the first phase angle set (
) and the second phase angle set () Reproducing the first time history wind load (X(t)) and the second time history wind load (Y(t)), respectively; A time history wind load calculation method considering correlation, characterized in that it consists of: - 제1항에서,In paragraph 1:상기 (a) 단계는 풍동실험 데이터로부터 위상차의 코사인 값에 대한 앙상블 평균() 데이터를 제공받아 상기 위상차의 코사인 값에 대한 앙상블 평균에 의해 위상차의 앙상블 평균(
)을 아래 [수학식] 에 의해 가정하는 것을 특징으로 하는 상관성을 고려한 시간이력풍하중 산정 방법.The step (a) is the ensemble average of the cosine value of the phase difference from the wind tunnel experiment data (
) Data is provided and the ensemble average of the phase difference is calculated by the ensemble average of the cosine value of the phase difference () is a time history wind load calculation method considering correlation, characterized by assuming the following [Equation].[수학식][Equation]
- 제2항에서,In paragraph 2,상기 (c) 단계는 아래 [수학식] 에 의해 앙상블 평균()을 보정하는 것을 특징으로 하는 상관성을 고려한 시간이력풍하중 산정 방법.Step (c) above is performed by the ensemble average (
) Time history wind load calculation method considering correlation, characterized by correcting.[수학식][Equation]
여기서 좌변의는 보정된 앙상블 평균, Here on the left side
is the corrected ensemble mean,보정계수,
correction coefficient
, - 제3항에서,In paragraph 3,상기 (b) 단계는 아래 [수학식] 에 의해 앙상블 표준편차()를 산정하는 것을 특징으로 하는 상관성을 고려한 시간이력풍하중 산정 방법.In step (b), the ensemble standard deviation (
) A time history wind load calculation method considering correlation, characterized by calculating.[수학식][Equation]
여기서,
, here
,,는 위상차의 코사인의 앙상블 표준편차.
is the ensemble standard deviation of the cosine of the phase difference.
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