In the design of structural wind resistance, it is necessary to consider the combination of load effect extremum caused by each wind component. The existing combination rules for Gaussian wind load effects are not applicable to the combination of non-Gaussian wind load effects. The Hermite polynomial transformation model is employed to transform the non-Gaussian wind effect process based on a potential standard Gaussian process in this paper. The probability distributions of the non-Gaussian wind effect process and the non-Gaussian peak factor are deduced. A simplified TR1 (Turkstra) combination equation for the two-component non-Gaussian wind effect process and a numerical integration expression for the TR2 combination rule are proposed. The improved simplified CQC (complete quadratic combination) equations for two-component non-zero-mean softening and hardening non-Gaussian wind effect processes are derived. The accuracy and validity of these simplified combination equations are verified using the Monte Carlo simulation method. The downwind force, crosswind force, vertical lift force, downwind overturning moment, crosswind overturning moment and torsional moment are obtained by integrating the surface wind pressure of the building under wind load. However, the maximum effect caused by each component of wind load may not occur simultaneously. Therefore, in estimating the total maximum wind load effect of a building structure in wind-resistant design, the combination of these wind component effects should be considered. The combination of extrema of wind load components is crucial in structural wind-resistant design. The combination of wind load components in different directions should be considered not only for high-rise buildings but also for mid-rise and low-rise buildings. Tamura et al. [ 1] pointed out that when the downwind wind load component is considered in structural design, the peak normal stress for low-rise and mid-rise building columns are underestimated by nearly 30% on average. The study of load combination has always been a hot topic in the engineering field. Turkstra and Madsen [ 2] established the famous Turkstra combination method. However, this method cannot effectively reflect the possibility of simultaneous occurrence of load extrema, and the estimated combined load may be underestimated. Ferry Borges and Castanheta [ 3] established the FBC combination method by hypothesizing and simplifying the stochastic process of loads. Nevertheless, they ignored the short-term time-domain fluctuations and correlations of loads. The load is regarded as a sequence of independent random variables occurring in equal periods. The International Joint Commission on Structural Reliability (JCSS) published “Common Uniform Rules for All Types of Structures and All Types of Materials” in 1976 and proposed the load combination rule, which is the JCSS combination method. These load effect combination rules are applicable to the linear superposition of multiple independent loads. The combination rules are empirical in nature and lack a strict theoretical basis. However, the wind loads acting on different surfaces of the building and the resulting responses are correlated, and these combination methods cannot take into account the correlation of wind load responses. At the same time, the reliability level of the maximum load combination cannot be calculated quantitatively. In order to consider the random occurrence time, random duration, and random intensity of loads, Wen [ 4, 5] established L.C. (Load Coincidence method) based on the Poisson model, conducted a series of studies around this method, and extended the scope of application of the method to general situations. The method is rigorous in theory but complicated in calculation. Larrabee and Cornell [ 6] established a point span rate method suitable for the linear combination of stationary, independently loaded random processes. Solari and Pagnini [ 7] proposed a combination method based on scalar responses, such as stress response, and ignored the correlation between wind loads. Asami [ 8, 9] proposed a combination method applicable to wind load components in any two directions based on the Solari&Pagnini method and spectral modal technology by considering the correlation between the pulsation responses. The Architectural Institution of Japan (AIJ) provisions for wind load combinations, covering downwind, transverse wind, and torsional directions, are formulated based on analytical results derived from the Asami method. Bartoli et al. [ 10] discussed the quasi-static combination of wind loads, using the Copula-function-based method to establish the joint probability distribution of extrema of load components. Based on the high-frequency balance test and the Copula Frank function, Yao Bo et al. [ 11] constructed a joint probability distribution function for the effects of two orthogonal directional components of wind loads on high-rise buildings and solved the combined coefficient of wind loads with a certain reliability level based on the stress criterion and the correlation between variables. However, the Copula function is an empirical function, and its validity and rationality need to be further tested. In the process of structural calculation and analysis, the SRSS and CQC combination rules are widely used to carry out load effect combinations. The SRSS combination rules are applicable to the extremum combination with a Gaussian process and no correlation of load effect, and the CQC combination rules are applicable to the extremum combination with a Gaussian process and correlation of load effect. For the edge of the building structure or the area with a sudden change in shape, due to the rapid change in the flow field, a vortex organization with strong spatial correlation is formed nearby, and the surface wind pressure in these areas often has obvious non-Gaussian characteristics. Thus, the accurate extremum of the wind effect cannot be obtained by using the CQC or SRSS combination rules. The reasonable combination of the extremum of non-Gaussian wind load effect is an important guarantee for structural wind-resistant design. For the non-Gaussian wind pressure process, a variety of conversion process models are proposed in existing studies [ 17, 18, 19, 20]. The non-Gaussian distributed wind pressure process can be expressed as a nonlinear polynomial combination form of the Gaussian process by implicit transformation [ 21, 22] or explicit transformation (1987, 2015) [ 23, 24], and a one-to-one mapping relationship between the non-Gaussian distributed wind pressure and the Gaussian process can be established. Winterstein et al. [ 25, 26] and Yang et al. [ 27] have successively proposed several sets of empirical approximate formulas for calculating polynomial coefficients from target skewness and kurtosis values. The Hermite polynomial model has a certain range of applications to non-Gaussian distributions. Based on the research of Winterstein et al., in order to further expand the scope of application, scholars have proposed some improved methods for hardening and softening processes, respectively. For the hardening process, Winterstein et al. [ 26] modified the transformation formula and proposed a Hermite polynomial model for it. Based on this work, Choi and Sweetman [ 28] proposed a semi-analytic formula to solve the parameters of the Hermite polynomial model for the hardening process. Chen and Huang [ 29] analyzed the peak response within a given time based on the crossing rate theory of non-Gaussian processes and vector-valued Gaussian processes, and they proposed a non-Gaussian response vector combination rule. Based on the research of Chen and Huang, Gong and Chen [ 30] discussed the calculation of the scalar and vector combination extremum of two non-Gaussian characteristic response components and the component extremum combination rule, targeting two response components with zero mean response. In this paper, the Hermite polynomial transformation model is employed to transform the softening, hardening and skewed non-Gaussian processes based on the potential standard Gaussian processes. The probability distribution of the non-Gaussian wind effect process is derived, followed by the presentation of the probability distribution of the non-Gaussian wind effect peak factor. Based on the POD (Proper Orthogonal Decomposition) of correlation coefficients of the non-Gaussian wind effect process and the probability distribution of the function, the joint probability density of the normalized softening, hardening and skew non-Gaussian wind effect process is derived. The TR rule, also known as the adjoint rule, combines the maximum value of one wind effect component with the instantaneous values of the remaining components, one by one, and takes the maximum of these combined values as the extreme value of the TR rule combination. The TR rule assumes that the extreme value of the combined response occurs simultaneously with the extreme values of the individual components. Theoretically, the probability distribution function of the extreme value of the TR rule combination cannot be obtained explicitly; however, its upper and lower bounds can be determined. Based on these bounds, approximate combination values are derived, referred to as the TR1 and TR2 combination rules, respectively. The TR rule is an empirical combination method, and previous studies on the TR combination rules have not approached it from a probabilistic perspective. Based on probability distribution theory, a simplified formula for the TR1 combination rule and the numerical integration expression of the TR2 combination rule are put forward. The improved CQC combination formulas for the maximum and minimum process of the softening and hardening non-Gaussian wind effect with non-zero means are proposed, along with the simplified combination coefficient expressions. The Monte Carlo simulation method is utilized to verify the accuracy and effectiveness of the proposed simplified combination formulas. 2. Methods To study the bound crossing rate and the extremum distribution of the stationary non-Gaussian process, Grigoriu [ 31] considered the stationary non-Gaussian process as a nonlinear function of the standard stationary Gaussian process. He then established a monotonic mapping relationship between the non-Gaussian and Gaussian processes. When the standard Gaussian process exceeded the bound, the non-Gaussian process also exceeded the bound. Thus, the extremum of a stationary non-Gaussian process can be expressed as a nonlinear function of the extremum of a standard Gaussian process. Applying the aforementioned theory to the non-Gaussian process, we can get g N ( t ) = f g ( t ) . (1) where f ⋅ represents a nonlinear transformation function; g ( t ) represents a standard stationary Gaussian process; and the subscript N denotes the non-Gaussian process. g ( t ) can be expressed in the form of an inverse function, i.e., g ( t ) = f − 1 g N ( t ) (2) where f − 1 ⋅ represents the nonlinear inverse transformation function. The Hermite polynomial transformation based on moments establishes a relationship between the standard non-Gaussian process and the standard Gaussian process through a monotonic conversion function. When the kurtosis coefficient of non-Gaussian process g N ( t ) is m 4 ≥ 3 , g N ( t ) is called the softening process; when the kurtosis coefficient of the non-Gaussian process is m 4 0 (13) or g N ( t ) ≤ − k h 3 + 1 / 4 h 3 , h 3 < 0 (14) 3. Theoretical Developments R 1 t , R 2 t , … , R n t are the non-Gaussian response process components with correlation, and the total response is as follows. R ( t ) = ∑ i = 1 n R ପ୍ତ i + ∑ i = 1 n R ˜ i ( t ) (15) where R ପ୍ତ i is the mean value of the process component of the i th non-Gaussian distributed response; R ˜ i ( t ) is the pulsation value of the i th non-Gaussian distributed response process component. As stated in Section 2, the non-Gaussian distributed response process can be represented as a monotonic function of the standard Gaussian process through Hermite polynomials. Based on the probability distribution function and probability distribution function of the standard Gaussian distribution response history, along with the corresponding Hermite transform polynomial, the expressions for the probability distribution function and probability density function of the non-Gaussian distribution response history can be derived. 3.1. Probability Distribution of Non-Gaussian Wind Effect Process Extrema 3.1.1. Softening Process According to Equation (3), the softening non-Gaussian process R i ( t ) is expressed as Equation (16) using the Hermite polynomial transform based on the standard Gaussian process u i ( t ) . R i t = R i ପ୍ତ + σ i ⋅ k u i t + h 3 u i 2 t − 1 + h 4 u i 3 t − 3 u i t (16) where σ i is the standard deviation of R i ( t ) . A relationship as described below exists between the probability density function of the non-Gaussian wind effect process and that of the Gaussian wind effect process. f R i d r i = f u i d u i (17) The probability density function of the softening non-Gaussian wind effect process R i ( t ) can be obtained from Equations (16) and (17). f R i r i = 1 σ i ⋅ 1 k 1 + 2 h 3 u i + 3 h 4 u i 2 − 1 ⋅ 1 2 π ⋅ exp − u i 2 2 (18) Based on the Poisson hypothesis, the probability distribution function of the extremum of the standard Gaussian wind effect process can be represented by Equation (19). P ( u imax ) = exp − ν 0 + T ⋅ exp − u imax 2 2 (19) where v 0 + is the average positive crossing zero rate; T is the duration; and u i , max is the extremum of the i th Gaussian wind effect process. Given the probability distribution function of the extremum of the standard Gaussian wind effect process, the corresponding probability density function can be derived as follows. p ( u i , max ) = ν 0 + T ⋅ u i , max ⋅ exp − u i , max 2 2 − ν 0 + T ⋅ exp − u i , max 2 2 (20) From Equations (17) and (20), the probability density of the extremum of the softening non-Gaussian wind effect process R i ( t ) is as follows. f ( R i , max ) ( r i ) = 1 σ i ⋅ 1 k 1 + 2 h 3 u i , max + 3 h 4 u i , max 2 − 1 ν 0 + T ⋅ u i , max ⋅ exp − u i , max 2 2 − ν 0 + T ⋅ exp − u i , max 2 2 . (21) Given the standard deviation, skewness, kurtosis, positive crossing zero rate and duration of the softening non-Gaussian wind effect process, the probability density and probability distribution of the extremum of the softening non-Gaussian process can be determined. The probability density curve of the extremum of the softening non-Gaussian process at skewness m 3 = 0 , kurtosis m 4 = 5 , v 0 + = 1.67 , T = 600 s , and σ i = 1 is shown in a, and the probability distribution curve of the extremum is shown in b. 3.1.2. Hardening Process According to Equation (7), the hardening non-Gaussian wind effect process R i ( t ) and the corresponding Gaussian wind effect process u i ( t ) have the following implicit expression. u i ( t ) R i ( t ) − R ପ୍ତ i σ i − h 3 R i ( t ) − R ପ୍ତ i σ i 2 − 1 − h 4 R i ( t ) − R ପ୍ତ i σ i 3 − 3 R i ( t ) − R ପ୍ତ i σ i (22) According to Equations (17) and (22), the probability density function of the hardening non-Gaussian wind effect process is as follows. f R i ( r i ) = 1 2 π ⋅ σ u i ⋅ exp − u i 2 2 σ u i 2 ⋅ 1 σ i ⋅ 1 − 2 h 3 ⋅ r i − r ପ୍ତ i σ i − 3 h 4 r i − r ପ୍ତ i σ i 2 − 1 (23) The probability density of the extremum of the hardening non-Gaussian wind effect process is as follows. f R i , max ( r i ) = ν 0 + T ⋅ u i , max ⋅ exp − u i , max 2 2 − ν 0 + T ⋅ exp − u i , max 2 2 ⋅ 1 σ i ⋅ 1 − 2 h 3 ⋅ r i − r ପ୍ତ i σ i − 3 h 4 r i − r ପ୍ତ i σ i 2 − 1 (24) The probability density curve of the extremum of the hardening non-Gaussian process at skewness m 3 = 1 , kurtosis m 4 = 2 , v 0 + = 2 , T = 600 s , and σ i = 1 is shown in c, and the probability distribution curve of the extremum is shown in d. 3.1.3. Skew Process Based on the transformation Equation (11) between the skewed process and the standard Gaussian process, as well as the probability density function of the extremum of the standard Gaussian process given in Equation (20), the probability density function of the extremum of the skewed process can be deduced. f ( R i , max ) ( r i ) = 1 σ i ⋅ 1 k 1 + 2 h 3 g ( t ) ⋅ ν 0 + T ⋅ u i , max ⋅ exp − u i , max 2 2 − ν 0 + T ⋅ exp − u i , max 2 2 (25) Given the standard deviation, skewness, kurtosis, positive zero-crossing rate, and duration of the skewed non-Gaussian wind effect process, one can determine the probability density and distribution of its extremum. The probability density curve of the extremum of the skewed non-Gaussian process at skewness m 3 = 1 , kurtosis m 4 = 3 , v 0 + = 2 , T = 600 s , and σ i = 1 is shown in e, and the probability distribution curve of the extremum is shown in f. 3.2. Probability Distribution of Non-Gaussian Wind Effect Peak Factor The Hermite polynomial transformation establishes a one-to-one correspondence between the normalized non-Gaussian wind effect extrema and the standard Gaussian wind effect extrema. Based on this correspondence and the probability distribution of the standard Gaussian wind effect extrema, the probability distribution of the normalized non-Gaussian wind effect extrema can be derived. Furthermore, the peak factor of the non-Gaussian wind effect for a specified occurrence probability can be obtained from the probability distribution of the non-Gaussian wind effect’s extrema. In this section, the probability distributions of the peak factor of the non-Gaussian wind effect are presented according to the Hermite polynomial transformation equations of the softening and hardening processes, respectively. The proposed formulas are applicable to wind load time histories that are stationary stochastic processes, and deviations from this assumption would make them inapplicable. 3.2.1. Normalized Softening Non-Gaussian Process Peak Factor u ( t ) is a standard Gaussian process with a mean of zero and a variance of 1. Assuming that the mean of R ( t ) is zero and the variance is 1, Equation (26) is obtained from the transformation of Equation (3): R = k u + h 3 u 2 − 1 + h 4 u 3 − 3 u (26) The derivative of R ( t ) is as follows. R ⋅ = k u ˙ 1 + 2 h 3 u + 3 h 4 u 2 − 1 (27) R ˙ has a mean of zero, and u ˙ also follows a Gaussian distribution, with a mean of zero and a variance of σ u ˙ 2 , and u ˙ and u are independent of each other. The variance σ r ˙ 2 of R ˙ is Equation (28). E R ˙ 2 = k 2 ⋅ σ u ˙ 2 ⋅ 1 + 4 h 3 2 + 18 h 4 2 (28) From Equation (28), we get Equation (29). σ u ˙ 2 = σ r ˙ 2 k 2 1 + 4 h 3 2 + 18 h 4 2 (29) The positive crossing zero rate of the extremum of the standard Gaussian process u ( t ) is as follows. ν 0 , u + ≈ 1 2 π ⋅ ∫ − ∞ + ∞ ω 2 S r ω d ω (30) where S r ( ω ) is the power spectral density of the Fourier transform of R ( t ) ; ω is the frequency corresponding to S r ( ω ) . When the mean of the non-Gaussian wind effect process is zero and its variance is unity, the maximum value is equivalent to the positive peak factor. The probability distribution of the positive peak factor for the softening non-Gaussian process is presented as follows. F R r = 1 k 1 + 2 h 3 u + 3 h 4 u 2 − 1 v 0 , u + T ⋅ u ⋅ exp − u 2 2 − v 0 , u + T ⋅ exp − u 2 2 (31) The mean value of the positive peak factor of the softening non-Gaussian process is as follows. E r = ∫ 0 + ∞ k u + h 3 u 2 − 1 + h 4 u 3 − 3 u ⋅ v 0 , u + T ⋅ u ⋅ e x p [ − ( u 2 ) / 2 ] exp − ν 0 , u + T exp − u 2 2 d u (32) Equation (32) represents the mean value of the positive peak factor, whereas Equation (33) represents the mean value of the negative peak factor. E − r = ∫ − ∞ 0 k − u + h 3 u 2 − 1 − h 4 u 3 − 3 u ⋅ v 0 , u + T ⋅ u ⋅ e x p [ − ( u 2 ) / 2 ] exp − ν 0 , u + T exp − u 2 2 d u (33) 3.2.2. Hardening Non-Gaussian Process Peak Factor Based on Equation (7), the implicit transformation equation for the hardening non-Gaussian process is as follows. u = r − h 3 r 2 − 1 − h 4 r 3 − 3 r (34) From the derivative of the expression (14) for the extremum distribution of the standard Gaussian process, Equation (17) and Equation (34), the probability density of the peak factor of the hardening non-Gaussian process is as follows. f R r = v 0 , u + T ⋅ u ⋅ exp − u 2 2 − v 0 , u + T ⋅ exp − u 2 2 ⋅ 1 − 2 h 3 r − 3 h 4 r 2 − 1 . (35) The mean of the positive peak factor associated with the hardening non-Gaussian process is presented as follows. E R = ∫ 0 + ∞ r f u max u d u (36) The mean of the negative peak factor corresponding to the hardening non-Gaussian process is presented as follows. E [ ( − R ) ] = ∫ − ∞ 0 ( − r ) f u max ( u ) d u (37) By substituting Equations (10) and (20) into Equations (36) and (37), respectively, the mean values of the positive and negative peak factors of the hardening non-Gaussian process can be derived. 3.3. Joint Probability Density of Non-Gaussian Wind Effect Components In this section, the correlation coefficient matrix among the components of the non-Gaussian wind effect is decomposed using POD. Subsequently, the non-Gaussian wind effect process is represented as a function of the non-Gaussian normalized basis vector. Based on the Hermite transformation, the basis vectors of the softening, hardening, and skewed non-Gaussian wind effects are represented as polynomials of the corresponding basis vectors of the Gaussian wind effect. By leveraging the probability density relationship between the non-Gaussian process and the Gaussian process, the probability density of the normalized basis vector of the non-Gaussian wind effect process can be derived. This probability density is a function of the probability density of the standard Gaussian process. Consequently, the joint probability density of the normalized basis vectors of the non-Gaussian wind effect process can be deduced. The joint probability density of the normalized non-Gaussian process can be inferred from the functional relationship between the non-Gaussian wind effect process and the corresponding normalized basis vector, along with the determinant of the Jacobian matrix. The non-Gaussian wind effect components are denoted as R ( t ) = R 1 ( t ) , R 2 ( t ) , ⋯ , R n ( t ) T , which can be expressed as the following equation. R ( t ) = R ପ୍ତ + σ R R ˜ ( t ) . (38) where σ R is the standard deviation of the non-Gaussian wind effect; R ପ୍ତ is the mean of the non-Gaussian wind effect process; and R ˜ ( t ) is the normalized pulsation process of the non-Gaussian wind effect. The correlation coefficient matrix of R ˜ ( t ) is decomposed by POD: R ˜ ( t ) = ψ σ V V t . (39) where ψ is the eigenvector of the correlation coefficient matrix of R ˜ ( t ) , σ V = d i a g λ 1 , λ 2 , ⋯ , λ n is the standard deviation of the base vector, and V ( t ) is the normalized base vector. The normalized basis vector expression is as follows. V 1 ( t ) V 2 ( t ) ⋮ V n ( t ) = 1 / λ 1 1 / λ 2 ⋱ 1 / λ n ψ 11 ψ 21 … ψ n 1 ψ 12 ψ 22 … ψ n 2 ⋮ ⋮ ⋱ ⋮ ψ 1 n ψ 2 n … ψ n n R ˜ 1 ( t ) R ˜ 2 ( t ) ⋮ R ˜ n ( t ) . (40) From the probability density relation of the function, the joint probability density of components of the normalized pulsation process of the non-Gaussian wind effect R ˜ ( t ) is as follows. f R ˜ 1 ⋯ R ˜ n r ˜ 1 ⋯ r ˜ n = ∏ j = 1 n f V j v j ⋅ J (41) where f v 1 ⋯ v n ( v 1 ⋯ v n ) is the joint probability density of the normalized basis vector, and J is the determinant of the Jacobian matrix. According to the probability density relation f V j d V j = f U j d u j between the non-Gaussian process and the Gaussian process, the probability density of softening normalized basis vector V j ( t ) can be obtained. f V j ( v j ) = 1 1 + 2 h 3 u j + 3 h 4 ( u j 2 − 1 ) ⋅ 1 2 π ⋅ exp − u j 2 2 (42) When the kurtosis coefficient of the normalized basis vector V j ( t ) is greater than 0 and less than 3, according to Hermite transformation Equation (7), the normalized basis vector of the hardening non-Gaussian wind effect can be expressed as a function of the basis vector of the corresponding Gaussian process. Based on the probability density relationship between the non-Gaussian process and the Gaussian process, the probability density of the normalized basis vector V j ( t ) of the hardening non-Gaussian wind effect process is as follows. f V j v j = 1 + 2 h 3 v j + 3 h 4 v j 2 − 1 ⋅ 1 2 π ⋅ exp − u j 2 2 (43) According to the probability density relationship between the non-Gaussian process and the Gaussian process, the probability density of normalized basis vector V j ( t ) of the skew non-Gaussian wind effect process can be obtained. f V j v j = 1 1 + 2 h 3 u j ⋅ 1 2 π ⋅ exp − u j 2 2 (44) By substituting Equations (42), (43) and (44), respectively, into Equation (41), the joint probability densities of components of normalized softening, hardening and skewing non-Gaussian wind effects R ˜ ( t ) can be obtained. 4. Combination Rules of Non-Gaussian Wind Effect Components and Discussion In the wind-resistant design of building structures, it is necessary to multiply the extremum of each wind load effect component by a combination coefficient to obtain the extremum of the combined wind load effect. The accuracy of the combined extremum should meet engineering requirements. Thus, it is crucial to determine simple and reasonable combination rules for the extrema combination of wind load effect components. The Turkstra (TR), SRSS, and CQC rules for estimating the linear combinations of the extrema of wind load effects are applicable for Gaussian processes of wind load effects but not for non-Gaussian ones. In this section, we derive the probability distribution expressions for the combined extrema of non-Gaussian wind effect components according to the TR rule and its approximations. Additionally, we propose a simplified equation for the improved CQC combination rule for non-Gaussian wind effect components. The Monte Carlo simulation method is employed to analyze the accuracy of the improved CQC combination rule for non-Gaussian wind effect components. Generate a standard normal random number sequence with a mean of 0 and a variance of 1 using the randn function in MATLAB (R2023b). Then, mathematically process these two standard normal random number sequences in MATLAB to obtain two non-Gaussian random number sequences with specified correlation coefficients, mean, variance, skewness, and kurtosis. As a result, two columns of non-Gaussian wind load time histories with specified statistical feature parameters are generated, and the two non-Gaussian wind load time histories have a specified correlation coefficient between them. 4.1. TR Combination Rule The TR rule, also known as the adjoint rule, combines the maximum value of one wind effect component with the corresponding adjoint values of the remaining wind effect components sequentially. Then, it selects the largest combined value as the combined extremum. Assuming that the extremum of the combined response coincides with the extremum of a wind effect component, the extremum of the scalar sum of n non-Gaussian wind effect components can be represented by Equation (45). R max = max X ^ 1 , X ^ 2 , … X ^ i , … , X ^ n (45) where X ^ i is the combination effect: the i-th wind effect component takes the extremum and the other wind effect components take the adjoint value. 4.2. TR1 Combination Rule Based on the research by Naess et al. [ 32], the probability distribution function of the extrema of the scalar sum of wind effect components is related as follows. F R ≤ min F X ^ 1 , … , F X ^ i − 1 , F X ^ i , F X ^ i + 1 , … , F X ^ n (46) where F R and F X ^ i are the probability distribution function of the extrema of wind effect components scalar sum and the probability distribution function of X ^ i , respectively. The expression of the probability density function of X ^ i is f X i ( x i ) = ∫ − ∞ + ∞ … ∫ − ∞ + ∞ ∫ − ∞ + ∞ f ( R 1 , R 2 , R 3 , … , R n ) ( r 1 , x i − ( r 1 + r 3 + r 4 + … r n ) , r 3 , … , r i , r n ) f R i ( r i ) ⋅ f R i , max ( r i ) d r 1 d r 3 d r 4 … d r n (47) The conditional probability density function of R 2 ( t ) when R 1 ( t ) takes the maximum value is: f R 2 R 1 ( r 2 ) = 1 σ 2 ⋅ 1 k ( 1 + 2 h 3 , 2 u 2 + 3 h 4 , 2 ( u 2 2 − 1 ) ) ⋅ 1 2 π 1 − ρ 2 ⋅ exp [ − 1 2 ( 1 − ρ 2 ) ⋅ ( u 2 − ρ u 1 , max ) 2 ] (48) where h 32 is the third Hermite moment of the second non-Gaussian wind effect component; h 42 is the Hermite fourth moment of the second non-Gaussian wind effect component. Similarly, when R 2 ( t ) takes its maximum value, the conditional probability density function of R 1 ( t ) is: f R 1 R 2 ( r 1 ) = 1 σ 1 ⋅ 1 k ( 1 + 2 h 3 , 1 u 1 + 3 h 4 , 1 ( u 1 2 − 1 ) ) ⋅ 1 2 π 1 − ρ 2 ⋅ exp [ − 1 2 ( 1 − ρ 2 ) ⋅ ( u 1 − ρ u 2 , max ) 2 ] (49) The expectation of the conditional probability of R 2 ( t ) when R 1 ( t ) takes its maximum is: E [ R 2 ( 1 ) ] = ∫ − ∞ + ∞ [ u 2 + h 3 , 2 ( u 2 2 − 1 ) + h 4 , 2 ( u 2 3 − 3 u 2 ) ] ⋅ 1 2 π 1 − ρ 2 ⋅ exp [ − 1 2 ( 1 − ρ 2 ) ⋅ ( u 2 − ρ u 1 , max ) 2 ] d u 2 (50) Similarly, the expectation of the conditional probability of R 1 ( t ) when R 2 ( t ) is at its maximum is: E [ R 1 ( 2 ) ] = ∫ − ∞ + ∞ [ u 1 + h 3 , 1 ( u 1 2 − 1 ) + h 4 , 1 ( u 1 3 − 3 u 1 ) ] ⋅ 1 2 π 1 − ρ 2 ⋅ exp [ − 1 2 ( 1 − ρ 2 ) ⋅ ( u 1 − ρ u 2 , max ) 2 ] d u 1 (51) When the non-Gaussian wind effect components are combined according to the TR1 rule, the average value of the lower bound of the extremum is E [ R max ] = max r 1 , max + E R 2 1 , r 2 , max + E R 1 2 (52) By substituting Equations (50) and (51) into Equation (52), the extremum of the combination of two non-Gaussian wind effect components can be derived according to the TR1 rule. 4.3. TR2 Combination Rule Based on the TR rule, the combination extremum obtained from the lower bound of the probability distribution is the combination value of the TR2 rule. When there are two wind effect components, the limit of the probability of the combined extremum is represented by Equation (53). f R max ( x ) = f X ^ 1 ( x 1 ) ⋅ F X ^ 2 ( x 2 ) + F X ^ 1 ( x 1 ) ⋅ f X ^ 2 ( x 2 ) . (53) When R 1 ( t ) takes the maximum value and R 2 ( t ) takes the synchronous value, the probability density function of X ^ 1 is f X ^ 1 x 1 = ∫ 0 + ∞ f R 2 R 1 x 1 − r 1 ⋅ f R 1 , max r 1 d r 1 . (54) If the wind effect process is softening a non-Gaussian process, the extremum probability density when R 1 ( t ) is the maximum in Equation (55). f R 1 , max ( r 1 ) = 1 k 1 ( 1 + 2 h 3 , 1 u 1 + 3 h 4 , 1 ( u 1 2 − 1 ) ) v 0 , 1 + T ⋅ u 1 ⋅ exp [ − v 0 , 1 + T ⋅ exp ( − u 1 2 2 ) ] ⋅ exp [ − u 1 2 2 ] . (55) Similarly, when R 2 ( t ) takes the maximum value and R 1 ( t ) takes the synchronous value, the probability density function of X ^ 2 is f X ^ 2 x 2 = ∫ 0 + ∞ f R 1 R 2 x 2 − r 2 ⋅ f R 2 , max r 2 d r 2 (56) If the wind effect process is softening a non-Gaussian process, when R 2 ( t ) is the maximum, the extremum probability density is f R 2 , max ( r 2 ) = 1 k 2 ( 1 + 2 h 3 , 2 u 2 + 3 h 4 , 2 ( u 2 2 − 1 ) ) v 0 , 2 + T ⋅ u 2 ⋅ exp [ − v 0 , 2 + T ⋅ exp ( − u 2 2 2 ) ] ⋅ exp [ − u 2 2 2 ] . (57) The limit of the mean value of the combination extremum of two softening non-Gaussian wind effect components combined according to the TR2 rule is R ପ୍ତ max ≈ ∫ − ∞ + ∞ x f R max x d x (58) Substituting Equations (48) and (55) into Equation (54) gives f X ^ 1 x 1 , and substituting Equations (49) and (57) into Equation (56) gives f X ^ 2 x 2 . Finally, by substituting f R max x into Equation (58) and integrating them numerically, the extremum average value of the combination of two softening non-Gaussian wind effect components according to the TR2 rule can be obtained. Based on the Hermite transformation equations corresponding to the hardening and skewed non-Gaussian wind effect processes, the expressions of the TR2 combination rule for these processes can be derived. This derivation is achieved by transforming the extremum probability density expressions of Equations (55) and (57). 4.4. MCQC Combination Rule The combined extremum of the CQC rule is equal to the square root of the complete quadratic extremum of each wind effect component. Moreover, the CQC rule can determine a more accurate combination value for the process components of the Gaussian-distributed wind effect. The CQC combination rule is applicable to Gaussian-distributed processes. In contrast, for non-Gaussian-distributed processes, they may be overestimated or underestimated, and the accuracy of the rule is related to the non-Gaussian characteristics of the wind effect process components. In this section, the improved parameters are introduced into the CQC combination rule. Subsequently, the maximum and minimum MCQC combination equations for the softening and hardening processes of non-Gaussian wind effects with non-zero means are deduced, respectively, and the simplified combination coefficients are proposed. Finally, the Monte Carlo simulation method is used to verify the applicability of MCQC combination rules for non-Gaussian wind effect components. 4.4.1. Softening Process R 1 ( t ) and R 2 ( t ) are the process components of the softening non-Gaussian distribution wind effect, the correlation coefficient is ρ , and the total response is R ( t ) = R ପ୍ତ 1 + R ପ୍ତ 2 + R ˜ 1 ( t ) + R ˜ 2 ( t ) (59) where R ପ୍ତ 1 and R ପ୍ତ 2 are the mean values of R 1 ( t ) and R 2 ( t ) , respectively; R ˜ 1 ( t ) and R ˜ 2 ( t ) R ˜ 2 ( t ) are the pulsation values of R 1 ( t ) and R 2 ( t ) , respectively. The absolute value of the maximum of the standard Gaussian process is equal to that of the minimum. Given the non-Gaussian characteristics of the non-Gaussian process and the non-linear transformation relationship between the non-Gaussian and Gaussian processes, it can be observed that the absolute values of the maximum and minimum of the non-Gaussian process are not equal. Therefore, the combination of the maximum and minimum of the non-Gaussian component process is studied. When two softening non-Gaussian wind effect components with non-zero mean are combined according to the CQC rule, the combination maximum value can be represented by Equation (60). R max = ( R 1 , max − g r 1 σ 1 ) + ( R 2 , max − g r 2 σ 2 ) + g r 1 2 σ 1 2 + 2 ρ g r 1 σ 1 g r 2 σ 2 + g r 2 2 σ 2 2 (60) where g r 1 and g r 2 are the positive peak factors of non-Gaussian process R 1 ( t ) and R 2 ( t ) , respectively, and their expressions are Equations (61) and (62) according to Hermite polynomial conversion. g r 1 = k 1 g N 1 + h 3 , 1 g N 1 2 − 1 + h 4 , 1 g N 1 3 − 3 g N 1 (61) g r 2 = k 2 g N 2 + h 3 , 2 g N 2 2 − 1 + h 4 , 2 g N 2 3 − 3 g N 2 (62) where g N 1 and g N 2 are the peak factors of non-Gaussian processes R 1 ( t ) and R 2 ( t ) correspond to Gaussian processes, respectively. shows the maximum combined value of two softening non-Gaussian wind effect components calculated according to the CQC rule, as well as the relative error between this combined value and the Monte Carlo simulation value. The error is presented as a function of the correlation coefficient and the peak factor ratio of the two components. As can be seen from , when two softening non-Gaussian wind effect components are combined according to the CQC rule, the relative error decreases exponentially with increasing correlation coefficient. When the correlation coefficient is less than 0.5, the relative error exceeds 10%. This indicates that the CQC rule is not suitable for combining non-Gaussian wind effect components, and thus, the CQC combination equation requires improvement. The characteristics of softening a non-Gaussian process are influenced by the skewness and kurtosis coefficient. By transforming Equation (60) and introducing improved parameters, the improved CQC combination equation for the maximum combination value of the two softening non-Gaussian wind effect components with non-zero mean is R max = R 1 , max + R 2 , max ⋅ φ 2 , max (63) where φ 2 , max is called the combination coefficient of R 2 , max , and its expression is: φ 2 , max = 1 − b 12 c 12 + 1 − ( b 12 c 12 ) 2 + 2 ρ b 12 c 12 + 1 R ପ୍ତ 2 g r 2 σ 2 + 1 ⋅ exp − 0.06 ୍ଠ ( 1 − ρ ) − 0.08 ୍ଠ ( b 12 − 0.6 ) + 0.06 ୍ଠ ( c 12 − 1 ) / c 12 . (64) where c 12 = σ 1 σ 2 , b 12 = g r 1 g r 2 . A Gaussian process with specified characteristics is generated using the Monte Carlo simulation method. Then, a softening non-Gaussian wind effect process with the same specified characteristics is obtained using Equation (3) based on the Hermite polynomial transformation. The combination values of two softening non-Gaussian wind effects are calculated using the MCQC combination Equation (63). The applicability of the MCQC combination equation is analyzed by comparing these combination values with the extrema of the Monte Carlo simulation process. When the skewness coefficients of one component of the two softening non-Gaussian wind effects are zero, positive, and negative, respectively, and the kurtosis coefficient of the other component remains constant, shows the relationship between the combined value obtained from the MCQC rule and the Monte Carlo simulated process value as a function of the kurtosis coefficient m41 and the correlation coefficient ρ. In the figures below, m41, m42, m31 and m32 are the kurtosis coefficient and skewness coefficient of components 1 and 2, respectively, and R MCQC and R MCQC are the MCQC combination value and Monte Carlo numerical simulation value, respectively. As shown in , except under complete negative correlation, the error between the combination result obtained by the MCQC rule and the Monte Carlo simulation value is less than 6%. The combination result obtained by the MCQC rule is in good agreement with the Monte Carlo simulation process. In structural design, when the wind effect components are completely negatively correlated, the combination value is small and may not be the critical value for the structure. When the kurtosis coefficients of the two components are equal, and the skewness coefficients of component 2 are zero, positive, and negative, respectively, shows the variation relationship between the combination result obtained by the MCQC rule and the Monte Carlo simulated process value as a function of the skewness coefficient m31 of component 1 and the correlation coefficient ρ. As shown in , it can be seen that when the skewness coefficient varies, the error between the combination value obtained from the MCQC rule and the Monte Carlo simulation process value is within 5%, except under complete negative correlation. From , it can be seen that the ratio of RMCQC/RMC is less than 1.0 when the correlation coefficient ρ = 1. This indicates that the combination value is smaller than the simulation value when the correlation coefficient ρ = 1. However, as shown in the figure, the maximum difference between the two values does not exceed 3%. Therefore, the combination value from the MCQC rule agrees well with the Monte Carlo simulation process value. The MCQC combination equation for the maximum value of two softening non-Gaussian wind effect components with non-zero mean can also be rewritten as Equation (65). R max = R 2 , max + R 1 , max ⋅ φ 1 , max (65) where φ 1 , max is called the combination coefficient of R 1 , max , and its expression is: φ 1 , max = 1 − 1 + b 21 c 21 − ( b 21 c 21 ) 2 + 2 ρ b 21 c 21 + 1 R ପ୍ତ 1 g r 1 σ 1 + 1 ⋅ exp − 0.06 ୍ଠ ( 1 − ρ ) − 0.08 ୍ଠ ( b 21 − 0.6 ) + 0.06 ୍ଠ ( c 21 − 1 ) / c 21 . (66) where c 21 = σ 2 σ 1 , b 21 = g r 2 g r 1 . The absolute values of the positive and negative peak factors of the standard Gaussian process are equal. In contrast, for the non-Gaussian process, the positive and negative peak factors differ, and so do the absolute values of its maximum and minimum pulsations. The minimum value of two softening non-Gaussian wind effect components with non-zero means, combined according to the CQC rule, can be represented by Equation (67). R min = R 1 , min − g ପ୍ତ r 1 σ 1 + R 2 , min − g ପ୍ତ r 2 σ 2 + g ପ୍ତ r 1 2 σ 1 2 + 2 ρ g ପ୍ତ r 1 σ 1 g ପ୍ତ r 2 σ 2 + g ପ୍ତ r 2 2 σ 2 2 (67) where g ପ୍ତ r 1 and g ପ୍ତ r 2 are the negative peak factors of the softening non-Gaussian processes R 1 ( t ) and R 2 ( t ) , respectively, and their expressions are Equations (68) and (65) according to Hermite polynomial conversion. g ପ୍ତ r 1 = − k 1 g N − h 3 , 1 g N 2 − 1 + h 4 , 1 g N 3 − 3 g N (68) g ପ୍ତ r 2 = − k 2 g N − h 3 , 2 g N 2 − 1 + h 4 , 2 g N 3 − 3 g N (69) Considering that the characteristics of softening non-Gaussian processes are influenced by the skewness coefficient and kurtosis coefficient, the MCQC combination equation for two softening non-Gaussian wind effect components with non-zero means can be obtained as Equation (70) by transforming Equation (67) and introducing the influence factor. R min = R 1 , min + R 2 , min ⋅ φ 2 , min (70) where φ 2 , min is called the combination coefficient of R 2 , min , and when the wind effect process components are positively correlated, uncorrelated, and negatively correlated, the expression of φ 2 , min is Equation (71), (72), and (73) respectively. φ 2 , min = 1 − b ପ୍ତ 12 c 12 + 1 − ( b ପ୍ତ 12 c 12 ) 2 + 2 ρ b ପ୍ତ 12 c 12 + 1 R ପ
