Moment-based low-rise building wind-induced vulnerability probability analysis method and system considering multiple disaster-inducing factors

Abstract

The invention relates to a moment-based low-rise building wind-induced vulnerability probability analysis method considering multiple disaster-inducing factors, and the method comprises the steps: firstly providing a low-rise building wind-induced vulnerability evaluation total probability framework considering directivity and multiple openings under the combined action of a flying object and wind pressure, and enabling the framework to consider directivity and multiple opening conditions at the same time; comprising the steps of calculating loss rates of windows / glass doors under different MRIs and calculating loss rates of roof panels under different MRIs; on the basis of the frame, a closed formula for determining the first eight-order original point moment of the loss rate and a display expression of the failure probability of the enclosure structure component are derived, and then estimated values of the loss rate of the enclosure structure component under different MRIs are obtained by adopting a maximum entropy model according to the original point moment; the invention further provides a moment-based high-efficiency probability analysis method for the wind-induced vulnerability of the low-rise building. The wind disaster assessment method has the double advantages of high precision and efficiency, the requirements of engineering and insurance industries can be better met, and the effectiveness of the assessment method is verified through numerical examples.

Description

Technical Field

[0001] The present invention belongs to the technical field of wind disaster assessment of low-rise building envelope structures, and relates to a method and system for probabilistic analysis of the wind-induced vulnerability of low-rise buildings taking into account multiple disaster factors. In particular, it relates to a moment-based, efficient and accurate method and system for probabilistic analysis of the wind-induced vulnerability of low-rise buildings taking into account directionality and multiple openings under the combined action of projectiles and wind pressure. Background Art

[0002] Low-rise buildings with wooden roof panels are widely used in residential communities due to their low cost and large living space. These structures are generally wind-sensitive and prone to severe damage in strong winds (Ellingwood et al., 2004; Gurley and Masters, 2011; Yang et al., 2018). Post-disaster investigations revealed that structural envelope components, such as windows, glass doors, and roof panels, were severely damaged by strong winds. This is attributed to the increased uplift forces on the roof panels after the failure of windows and glass doors on the windward wall (Habte et al., 2017; He et al., 2018; Lan and Huang, 2022; Sharma and Richards, 2005). It is worth noting that, in addition to strong wind pressure, the impact of wind-induced missiles is also a major factor in window and glass door failure (Abdelhady et al., 2020; Ginger et al., 2007; Pita et al., 2012). To meet the needs of the engineering and insurance industries and reduce wind damage, reliable and efficient wind damage assessment methods are needed.

[0003] Many researchers have used models based on wind damage data (Unanwa et al., 2000) or structural reliability theory (Ouyang and Spence, 2019; Sarma et al., 2023; Vickery et al., 2006a, b) to estimate wind-induced losses in roof panels. Compared to the former (i.e., data-based models), the latter (i.e., reliability-based models) (Ellingwood et al., 2004; Stewart et al., 2016; Zeng et al., 2021) may be more accurate for structures located in areas with insufficient hazard data and, therefore, have wider applications. Vickery et al. (2006a, 2006b) developed a wind damage and loss estimation model for low-rise buildings, which was used in the Hazardous Areas of the United States (HAZAUS-MH) model. Several researchers (Pinelli et al., 2011; Pita et al., 2012) have proposed public loss assessment models specifically designed to predict losses in residential buildings in Florida. These models are primarily used for loss estimation during hurricane events in the United States, involving simulations of hurricane wind fields and paths. Lee and Rosowsky (2005) analyzed the vulnerability of low-rise wood-frame roof panels based on code-determined wind loads. Using wind tunnel data, several studies (Huang et al., 2015; He et al., 2015; Stewart et al., 2016) effectively accounted for the uncertainty of pressure coefficients to assess damage to asphalt shingle roofs and metal roof sheathing. These models primarily aim to estimate wind-induced losses to low-rise building envelopes in specific directions without considering wind directionality. However, due to the significant directional effects of wind speed and aerodynamic forces, wind-induced losses to low-rise buildings vary significantly across different directions. Therefore, directionality should be considered in wind damage estimation. Ji et al. (2018) considered directionality in their wind-induced loss estimation for steel roofs. Wu et al. (2023a) proposed a novel wind damage estimation method that considers wind directionality, achieving high estimation efficiency. However, the extensive Monte Carlo simulations and multiple integrations make these methods computationally inefficient. Wu et al. (2023b) proposed a fully probabilistic framework that accounts for directionality and a moment-based method to more efficiently estimate wind-induced losses in roof panels under different mean return periods (MRIs). However, this method does not account for multiple openings in the windward wall caused by missiles, which may lead to an underestimation of wind damage losses. Summary of the Invention

[0004] In light of this, to address the problems of existing wind damage assessment methods, such as their biased estimates and low computational efficiency, which result in limited adaptability and an inability to meet the needs of the engineering and insurance industries, this present invention provides a moment-based probabilistic analysis method and system for wind-induced loss in low-rise buildings that considers multiple hazard factors. This analysis method derives the first eight moments of the loss rate and employs a maximum entropy model (MEM)-based approach to more effectively estimate wind-induced losses under different MRI conditions. This wind damage assessment method can be considered an enhanced version of existing probabilistic wind damage estimation methods in terms of estimation accuracy, helping to better meet the needs of the insurance industry. Numerical examples also verify the effectiveness of this assessment method.

[0005] In order to achieve the above object, the present invention provides the following technical solutions:

[0006] A moment-based method for analyzing the probability of wind-induced loss of low-rise buildings considering multiple hazard factors includes the following steps:

[0007] S1. Simplify the gradual damage of wind-induced disasters to the enclosure structures of low-rise buildings into a three-stage wind-induced disaster process;

[0008] S2. A fully probabilistic framework for wind-induced vulnerability assessment of low-rise buildings under the combined effects of missiles and wind pressure, taking into account directionality and multiple openings, is proposed. This framework considers both directionality and multiple openings, including the calculation of the loss rate of windows / glass doors under different mean re-intensity intervals (MRIs) and the calculation of the loss rate of roof panels under different mean re-intensity intervals (MRIs).

[0009] S3. Taking into account directionality and multiple openings, a closed-form formula for determining the first eight origin moments of the loss rate and an explicit expression for the failure probability of the envelope structure components are derived. Then, based on the origin moments, the maximum entropy model is used to obtain an estimated value of the loss rate of the envelope structure components under different MRIs, and an efficient analysis strategy for the wind-induced vulnerability of low-rise buildings is proposed.

[0010] The moment-based probabilistic analysis system for wind-induced vulnerability of low-rise buildings that considers multiple hazard factors includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, the above-mentioned probabilistic analysis method for wind-induced vulnerability of low-rise buildings is implemented.

[0011] The beneficial effects of the present invention are:

[0012] This invention discloses a moment-based probabilistic analysis method for the wind-induced vulnerability of low-rise buildings, considering multiple hazard factors. This method establishes a fully probabilistic analysis framework for the wind-induced vulnerability of low-rise buildings under the combined effects of missiles and wind pressure, taking into account directionality and multiple openings. By considering directionality and multiple openings, explicit expressions for determining the failure probabilities of multiple envelope components and a closed-form formula for calculating the first eight origin moments of the loss rate are derived. Based on the origin moments, a maximum entropy model is used to estimate the loss rate of envelope components under the mean regression period (MRI). This method proposes an efficient moment-based analysis strategy for the wind-induced vulnerability of low-rise buildings. This method is validated and demonstrated using a numerical example of a low-rise building based on a wind tunnel test. Numerical case analysis results show that the estimated loss rate of envelope components using this method agrees well with that based on the MCS method. Furthermore, compared to the MCS method, this method avoids numerous Monte Carlo simulations, significantly improving computational efficiency. This method offers high accuracy and computational efficiency, better meeting the needs of the engineering and insurance industries.

[0013] Other advantages, objectives, and features of the present invention will be described in the following description and, to some extent, will be apparent to those skilled in the art upon examination and study of the following, or may be taught from practice of the present invention. The objectives and other advantages of the present invention may be realized and obtained through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below with reference to the accompanying drawings, in which:

[0015] Figure 1 This is a simplified three-stage wind disaster process diagram;

[0016] Figure 2 It is the building configuration diagram in the specific numerical example;

[0017] Figure 3 is the layout diagram of the residential area in the specific numerical example;

[0018] Figure 4 For the specific numerical example, the AOA is 270° and the opening condition is =[1,1,1,0,0,0,1,1], the net wind pressure coefficient time history of roof panels A and B, where Figure 4 (a) is the time history comparison diagram of the net wind pressure coefficient and the external wind pressure coefficient of roof panel A. Figure 4 (b) is a time-course comparison diagram of the net wind pressure coefficient and the external wind pressure coefficient of roof panel B;

[0019] Figure 5 For the specific numerical example, the AOA is 270° and the opening condition is =[1,1,1,0,0,0,1,1], CDFs of the extreme net wind pressure coefficients of roof panels A and B, where Figure 5 (a) is a comparison chart of the extreme net wind pressure coefficient and the extreme external wind pressure coefficient CDFs of roof panel A. Figure 5 (b) is a comparison chart of the extreme net wind pressure coefficient and the extreme external wind pressure coefficient CDFs of roof panel B;

[0020] Figure 6 is the extreme wind pressure JCDF between roof panels A and B in the specific numerical example;

[0021] Figure 7 are the CDFs of the roof panel loss rate calculated using the proposed method in the specific numerical example, Figure 7 (a) = Probability distribution diagram of roof panel loss rate at 45°, Figure 7 (b) =90° roof panel loss rate probability distribution diagram, Figure 7 (c) =180° roof panel loss rate probability distribution diagram, Figure 7 (d) = Probability distribution of roof panel loss rate at 270°. DETAILED DESCRIPTION

[0022] The following describes the embodiments of the present invention through specific examples. Those skilled in the art will readily understand the other advantages and benefits of the present invention from the disclosure herein. The present invention may also be implemented or applied through various other specific embodiments, and the details in this specification may be modified or altered based on different viewpoints and applications without departing from the spirit of the present invention.

[0023] The moment-based probability analysis method for wind-induced loss of low-rise buildings considering multiple disaster factors includes the following steps:

[0024] S1. Simplify the gradual destruction process of low-rise building envelope structures;

[0025] Post-disaster investigations show that windows, glass doors, and roof panels are vulnerable components of low-rise building envelopes. Although the actual wind damage process is extremely complex, the main reasons can be summarized into two points: (1) The wind pressure on the envelope components generally exhibits non-Gaussian characteristics with large extreme values; (2) Strong wind pressure or wind-induced missile impact usually causes window / glass door failure (Barbato et al., 2013; FEMA 2009), resulting in a significant increase in internal pressure. Therefore, the uplift force on the roof panel increases significantly. Considering these important factors, our previous studies used a simplified progressive failure process (e.g., Wu et al., 2021, 2023a, 2024a) to estimate damage. This simplified process includes three stages, namely, stage 1: closed building, stage 2: partially closed building, and stage 3: roof panel loss. Assuming the building is an ideal closed structure, the first stage is negative internal pressure, and the second stage is a partially closed building with multiple openings on the wall and positive internal pressure. In the third stage, a plurality of openings are provided on the roof panel, so that the uplift force on the roof panel is smaller than that in the second stage. Figure 1 This process is presented in

[15] . This process considers the internal pressure changes caused by wall openings and their impact on the net wind pressure on the roof, as well as the worst-case scenario for roof wind damage, resulting in a relatively conservative wind damage estimate. A more detailed description of this process can be found in Wu et al. (2023a).

[0026] S2. Establish a full probabilistic assessment framework for wind-induced vulnerability of low-rise buildings under the combined effects of missiles and wind pressure, taking into account directionality and multiple openings;

[0027] A fully probabilistic framework for wind-induced vulnerability assessment of low-rise buildings under the combined action of missiles and wind pressure, considering directionality and multiple openings, is proposed. This framework takes both directionality and multiple openings into account, including the calculation of the loss rate of windows / glass doors under different mean refresh intervals (MRIs) and the calculation of the loss rate of roof panels under different mean refresh intervals (MRIs).

[0028] S21. Calculate the loss rate of windows / glass doors under different mean refresh intervals (MRI);

[0029] Usually, strong wind pressure or the impact effect of wind-induced missiles will cause the i-th (i 1, 2, …, M ) window / glass door failure (hereinafter referred to as WG). Given the limited data and the lack of basic research in this field, the proposed method system will temporarily exclude two potential correlations: (1) the correlation between the effects of strong wind pressure and wind-borne debris on the building envelope; (2) the spatial correlation of strong wind pressure or wind-borne debris effects that lead to correlated failures between different wall components. This simplification aims to maintain the feasibility of the analysis while leaving room for the theoretical framework to be improved with sufficient data. The i-th WG is the k-th ( 1, 2, …, ) direction of damage It can be expressed as:

[0030] (1)

[0031] in, and It represents the probability that the i-th WG is destroyed by the strong wind pressure and wind-induced missile impact in the k-th direction respectively.

[0032] S211. Determine the probability of WG damage caused by strong wind pressure;

[0033] Determine the limit state function of the i-th WG damage caused by strong wind pressure in the k-th direction It can be expressed as:

[0034] (2)

[0035] in, and denote the resistance and extreme wind pressure of the i-th WG in the k-th direction, respectively, and they are usually regarded as two independent random variables (e.g., Huang et al., 2016). and They represent the destruction and non-destruction of the i-th WG in the k-th direction, respectively.

[0036] Under the average wind speed in the kth direction, the extreme wind pressure of the i-th WG can be expressed as:

[0037] (3)

[0038] in, kg / m 3 is the air density; Indicates the i-th WG in the k-th direction The extreme value of represents the net wind pressure coefficient. Generally speaking, the transfer process method (e.g., Sadek and Simiu, 2002; Zhang et al., 2019) is recommended based on get The probability distribution of Subtract internal wind pressure coefficient ,Right now The internal pressure of stage 1 is usually caused by background leakage and is ignored here, so .

[0039] In order to estimate wind disaster losses more accurately, it is usually necessary to consider the uncertainty associated with the WG extreme wind pressure coefficient. Extreme wind pressure The cumulative distribution function (CDF) of can be expressed as:

[0040] (4)

[0041] in, Indicates Conditional Conditional CDF of (i=1, 2, …, M); (i=1, 2, …, M) indicates a given Reaching wind pressure level Time , obtained from formula (3).

[0042] according to , the probability of destruction and non-destruction of the i-th WG in the k-th direction and Calculated by the following equation

[0043] (5)

[0044] (6)

[0045] in, yes The probability density function (PDF) of is usually considered to follow a normal distribution or a lognormal distribution (e.g., Stewart et al., 2016).

[0046] S212. Determine the probability of WG damage caused by wind-induced missiles

[0047] Strong winds can carry upstream debris such as branches, stones, and tiles to impact the surface of a building, causing damage to vulnerable parts (such as windows / glass doors). These fragments are usually classified into compact, sheet-shaped, and rod-shaped. Generally speaking, it is difficult to establish a wind-induced missile hazard model that takes into account the initial conditions, flight trajectory, and impact on buildings of wind-induced missiles. Currently, the wind-induced missile hazard model proposed by Lin and Vanmarcke (2010) is widely used in wind hazard assessment and analysis of engineering structures. Based on this model, the probability that a wind-induced missile in the kth direction causes damage to the i-th WG is It can be expressed as:

[0048] (7)

[0049] in, is the area ratio between the WG and its surrounding walls; is the number of type d (d = 1, 2, …, D) fragments generated from the u (u = 1, 2, …, U) source building in the k direction (hereinafter referred to as type ud fragments); ,in Indicates the area of the target building projected onto the ground; is the probability density function value of the udth type of debris in the kth direction falling on the center of the target building; represents the probability that the impact momentum of the udth type fragment in the kth direction exceeds the impact resistance of the ith WG. The detailed estimation method of the parameters in formula (7) can be found in Ji et al. (2020). Obviously, determining these parameters usually requires building information and wind conditions.

[0050] S213. Determine the loss rate of WG under different MRI conditions

[0051] Indicates whether the i-th WG is damaged in the k-th direction, representing the damage situation The probability of occurrence is ,Right now , representing the undamaged situation The probability of occurrence is ,Right now ; Vector Used to represent the potential opening conditions in the kth direction, the total number of which is 2 M , among which indivual( 1, 2, …, 2 M )For opening working condition express;

[0052] The loss rate of WG in the kth direction can be expressed as:

[0053] (8)

[0054] Assuming that different WG failures are independent of each other (Ji et al., 2020), the probability of occurrence of each opening condition in the kth direction is Expressed as:

[0055] (9)

[0056] Based on formulas (8) and (9), the corresponding The CDF of can be expressed as:

[0057] (10)

[0058] in, , ( 1, 2, …, M-1) are variables equal to 0 or 1. Then, the CDF of the loss rate of the WG considering the directionality can be derived as:

[0059] (11)

[0060] in, Therefore (k=1, 2, …, K) is the condition Conditional CDF of yes JPDF.

[0061] along with Get, R year MRI (denoted as ), which can be calculated as:

[0062] (12)

[0063] in, yes The inverse function of .

[0064] S22. Calculate the loss rate of roof panels under different mean regression periods (MRI)

[0065] S221. Joint probability distribution of extreme wind pressure

[0066] according to , changes in external wind pressure at the opening will lead to changes in the internal pressure of the building. Previous studies (e.g., Holmes 1979; Oh et al., 2007) have shown that many factors, including building volume, opening location and number, will affect the internal pressure. In order to effectively consider the complexity and variability of opening conditions, the Bernoulli equation (e.g., Wu et al., 2024a) is usually used to simulate the internal pressure, and its simulation accuracy and efficiency are satisfactory. This study also uses the Bernoulli equation to simulate the internal pressure coefficient in the kth direction. For more information, please refer to Wu et al. (2024a). Get After that, the jth net wind pressure coefficient in the kth direction (j=1, 2, …, N) can be obtained by Get, among them is the jth external wind pressure coefficient in the kth direction. The corresponding extreme net wind pressure coefficient can be Obtained using the same method as WG.

[0067] Similar to formula (3), the average wind speed of the jth roof panel in the kth direction is Extreme wind pressure (k=1,2, …, K) can be determined as:

[0068] (13)

[0069] When considering the uncertainty of the extreme net wind pressure coefficient, the j-th roof panel with directionality and multiple openings is considered. The CDF of the extreme wind pressure ((j=1, 2, …, N)) can be expressed as:

[0070] (14)

[0071] in, Therefore and Conditional Conditional joint CDF (JCDF) for (k=1, 2, …, K); Therefore The JCDF of different opening conditions appears in each direction of the condition; yes JPDF for (k=1, 2, …, K).

[0072] The wind pressure coefficients in different directions are usually considered to be independent of each other (Tian and Chen, 2020), so formula (14) becomes:

[0073] (15)

[0074] in, Therefore and (k=1, 2, …, K) is the condition CDF of .

[0075] Assuming that the wind speed and opening conditions in each direction are also independent of each other, formula (15) can be further simplified as follows:

[0076] (16)

[0077] in, (k=1, 2, …, K) is the average wind speed in the kth direction Down Probability of occurrence; is the PDF of the annual extreme wind speed in the kth direction.

[0078] In addition, by generalizing formula (15) to multiple roof panels and taking into account the directionality and multiple openings, the JCDFs of multiple extreme wind pressures can be obtained as follows:

[0079] (17)

[0080] in, is the kth direction and The conditional JCDF of multiple extreme net wind pressure coefficients under the condition. Obviously, the multiple integrals in Equation (17) can be solved by the MCS method with satisfactory estimation efficiency. First, it can be obtained by get Multiple random samples (k=1, 2, …, K). Then, use these samples to calculate and . The ensemble mean of .

[0081] Furthermore, for multiple extreme net wind pressure coefficients under a specific wind direction, there are two special cases: independence and complete correlation. Therefore, formula (17) has two specific expressions. For the first case, when multiple extreme net wind pressure coefficients under the same wind speed direction are considered to be independent, the JCDF of multiple extreme wind pressures is expressed as:

[0082] (18)

[0083] in, Therefore and Conditional CDF of (j=1, 2, …,N; k=1, 2, …, K).

[0084] For the second case, when multiple extreme net wind pressure coefficients under the same wind speed are completely correlated, the JCDF of multiple extreme wind pressures is:

[0085] (19)

[0086] in, Therefore and CDF of the main extreme net wind pressure coefficient under the conditions.

[0087] When the wind speeds in different directions and the opening conditions are also independent of each other, it can be concluded that:

[0088] (20)

[0089] For the extreme wind pressure of a single roof panel obtained by equations (14)-(16), assuming that the directional annual extreme wind speed and the multiple extreme net wind pressure coefficients under the same directional wind speed are independent or completely correlated, the JCDFs of multiple extreme wind pressures can be easily obtained from their respective CDFs. For the case where the two are considered independent, the JCDF can be estimated as follows:

[0090] (twenty one)

[0091] For the case where both are considered perfectly correlated, the JCDF can be calculated as:

[0092] (twenty two)

[0093] in, is the CDF of the dominant extreme wind pressure.

[0094] S222, Evaluation of wind-induced damage to roof panels under different MRI conditions

[0095] In strong winds, whether the jth roof panel will fail depends on the resistance, uplift, and deadweight. For buildings with lightweight wooden frames, the deadweight load is usually ignored due to their light weight. In this study, the deadweight load is also not considered to obtain conservative results (Lee and Rosowsky, 2005). Therefore, the limit state function used to determine the failure of the jth roof panel caused by strong wind pressure in the kth direction is It can be given by the following formula:

[0096] (twenty three)

[0097] in, represents the resistance of the jth roof panel; represents the extreme wind pressure of the jth roof panel. (or ) indicates that the jth roof panel is damaged (or not damaged).

[0098] Considering the directionality and multiple openings, the jth roof panel failure and undamaged The probabilities can be expressed as:

[0099] (twenty four)

[0100] (25)

[0101] in, is the PDF of the resistance of the jth roof panel.

[0102] When there are N roof panels, the probability that all N roof panels are destroyed is and the probability that the jth (j = 1, 2,…, N-1) roof panel is damaged while the Nth roof panel is not damaged Can be expressed as:

[0103] (26)

[0104] (27)

[0105] in, is the JPDF of resistance, which is generally considered to be independent of each other (Wu et al., 2023b), that is, . Note that there are few common connections between log roof panels, which results in and The correlation between is very weak. Therefore, assuming that and Are independent of each other.

[0106] Similarly, use It represents the indicator variable, which indicates whether the j-th roof panel is damaged after considering all directions. (or 0) indicates damage (or no damage). Therefore, considering the directionality and multiple openings, the loss rate of the roof panel is expressed as:

[0107] (28)

[0108] Corresponding The CDF of can be expressed as:

[0109] (29)

[0110] in, , ( 1, 2, …, N-1) are variables equal to 0 or 1. Afterwards, R year MRI (denoted as ) can be calculated as:

[0111] (30)

[0112] in, yes The inverse function of .

[0113] S3. Obtain a wind-induced vulnerability analysis method considering directionality under the combined action of missiles and wind pressure based on the maximum entropy model;

[0114] For simplicity, we only analyze Calculation, calculation and Same. As mentioned above, The probability distribution of can be obtained by formula (29). However, there are multiple failure conditions in the calculation, and multiple integrations for each failure condition are very time-consuming. Therefore, this study proposes a moment-based method using the maximum entropy model (MEM) to obtain and This model is widely used to obtain CDF estimates of non-Gaussian variables. For details, see (Wang et al., 2021). In the developed MEM-based wind disaster assessment method, the origin moment of the loss rate is required. Once the origin moment is known, MEM can be used to estimate The PDF, i.e. . It is easy to get and Therefore, the following describes a method for estimating the original moments of the loss rate, including the derivation of formulas for determining the first eight original moments of the loss rate and the failure probability of the roof panel.

[0115] S31, calculation The closed-form formula for the origin moment of

[0116] Will Before ( =1, 2, ..., )-order origin moment is denoted as In order to achieve a relative balance between computational efficiency and accuracy, this study adopts =8. According to formula (28), It can be derived from the following formula:

[0117] (31)

[0118] (32)

[0119] (33)

[0120] (34)

[0121] (35)

[0122] (36)

[0123] (37)

[0124] (38)

[0125] It can be seen that before calculation The failure probability is required for the origin moment , , , , , , , ( , , , , , , , = 1, 2, …, N). These probabilities can be obtained from equations (24) and (26) based on different failure conditions. However, when N is large, these integral calculations are computationally inefficient. The following sections aim to develop corresponding analytical formulas to more efficiently obtain the roof panel failure probability.

[0126] S32. Estimation of roof panel failure probability

[0127] When the j-th roof panel considering the direction in formula (15) is obtained The CDF of is obtained as follows: When the CDF of is known. Then, based on formula (23), when assuming and irrelevant (e.g., Stewart et al., 2016), The CDF of can be derived as:

[0128] (39)

[0129] based on , through transfer process theory, and the corresponding standard Gaussian variable It can be expressed as:

[0130] (40)

[0131] in, is the transfer function; is based on The inverse CDF of . The inverse function corresponding to formula (40) is:

[0132] (41)

[0133] in, express The inverse function of .

[0134] Generally speaking, wooden roof panels and There is no correlation between ( , =1, 2, …, N). In addition, it is often assumed that the resistance of different roof panels is unrelated (e.g., Stewart et al., 2016), i.e. hour , hour .therefore, and Correlation coefficient It can be deduced as follows (Wu et al., 2023b):

[0135] (42)

[0136] in, yes and The correlation coefficient between .

[0137] according to , and Correlation coefficient It can be estimated by the following analytical formula:

[0138] (43)

[0139] in, is an analytical function with different forms in different cases. These analytical functions are summarized in Table 2 of Wu et al. (2024b) and are not listed here for simplicity. After determination, the failure probability , ,…, It can be calculated as:

[0140] (44)

[0141] (45)

[0142] (46)

[0143] (47)

[0144] (48)

[0145] (49)

[0146] (50)

[0147] in, express Joint PDF of dimensional standard Gaussian variables; ,…, is the covariance matrix. For the sake of brevity, this study only gives , and , as follows:

[0148] (51)

[0149] (52)

[0150] (53)

[0151] in, ( =1, 2, …, N; =1, 2, …, N).

[0152] Note that in formula (24) It can be expressed as:

[0153] (54)

[0154] Obviously, using equations (44)-(50) and (54) instead of equations (24) and (26) to calculate , , , ,…, and More convenient.

[0155] Specific numerical examples

[0156] 1. Select a building prototype and obtain wind tunnel test data

[0157] In order to verify the accuracy of the full probabilistic assessment framework for wind-induced damage considering directionality and multiple openings, and to evaluate the performance of the proposed MEM-based loss estimation method, this study analyzes a low-rise building with a roof slope of 1:12, such as Figure 2 The building is 19.2 meters long, 12.2 meters wide, and 3.7 meters high. There are 8 windows of 1.83 meters x 1.07 meters (width x height) symmetrically distributed on all the walls. Figure 2 In Chinese ~ The roof system consists of 80 wooden panels, secured to the frame with 6d standard nails. The wind pressure coefficient of the building envelope was determined through wind tunnel testing (Ho et al., 2005), and the relevant data is publicly available. Figure 2 The blue “+” in the figure indicates the pressure measurement points that record wind pressure. These pressure measurement points are available for various wind angles of attack (AOA), including AOA = 0°, 5°, 10°, …, 90° and AOA = 270°, 275°, 280°, …, 360°.

[0158] The moment-based probabilistic analysis method for wind-induced loss in low-rise buildings, which considers multiple hazards, indicates that the resistance of windows and roof panels, as well as the impact resistance of windows on windward walls, is required for wind damage estimation. The selection of these resistances is based on several studies (e.g., Lee and Rosowsky, 2005; Unikrishnan and Barbato, 2016; Wu et al., 2023b). The nominal values and coefficients of variation (COV) of their probability distribution models are summarized in Table 1. To account for resistance reductions caused by variations in materials, manufacturing, and response effects (e.g., Liu et al., 2019), the ratio of the mean to the nominal value was set to 0.88 in this study.

[0159] Table 1 Resistance of enclosure components

[0160]

[0161] 2. Analysis of wind-induced missile hazards

[0162] In order to consider the damage of windows caused by wind-induced missiles, a residential area layout is set up, such as Figure 3 As shown in Figure 1. In this residential complex, the building within the circle is the target building, and the other buildings are the source buildings, all of which are structurally identical to the target building. Accounting for all these factors is extremely complex, as damaged envelope components of upstream buildings (such as windows / glass doors, roof panels, and tiles) can serve as debris sources in a real-world disaster. This paper aims to demonstrate and illustrate the proposed method, which simplifies the sources of wind-borne debris, considering only failed roof panels as a type of debris (sheet-like). Using wind tunnel data, roof panel pressures are obtained and peak pressures are estimated using an interpolation technique based on proper orthogonal decomposition (POD) (e.g., Ji et al., 2018). Multiple simulations are performed, and peak pressures and resistance forces are compared to determine the number of damaged roof panels. Referring to Lin and Vanmarcke (2010), the wind-borne projectile hazard model uses Md = 4.08 kg, Ad = 0.143 m², and Tk = 1.58 s.

[0163] 3. Analysis of wind damage to roof panels

[0164] 3.1. Annual extreme wind speed by direction

[0165] Based on the moment-based probability analysis method for wind-induced loss of low-rise buildings, which considers multiple hazard factors, the joint probability distribution of directional extreme wind speeds was calculated. This study also used the wind speed data used in the numerical examples of Wu et al. (2023b). This data can be downloaded from the official website of the China Meteorological Administration. After processing the data using the same method as Wu et al. (2023b), 57 extreme wind speed samples were obtained for each of the eight directions: north, northeast, east, southeast, south, southwest, west, and northwest. The Gumbel distribution has been shown to be an excellent model for the probability distribution of wind speed samples in each direction (Bezabeh et al., 2018). Using a Gaussian copula, the joint probability distribution of the extreme wind speeds in each direction can be easily obtained. The determination process is described in Wu et al. (2023b).

[0166] 3.2 Extreme wind pressure

[0167] When estimating the extreme wind pressure on a roof panel, it is necessary to determine the probability of each opening condition occurring at a given wind speed in a given direction. Based on formula (1), the probability of a single WG being damaged at a specific wind speed in the kth direction can be estimated, and the probability of each opening condition can be calculated using formula (9). For simplicity, Table 2 only lists the opening conditions at AOAs of 270° and 360°. The four opening conditions with the highest probability of occurrence are respectively ( =1, 2, 3, 4). It can be seen that The range of variation is very large. This is because the internal pressure varies greatly under different opening conditions.

[0168] Table 2 AOA at 270° and 360° Four opening conditions

[0169]

[0170] For each opening condition , the internal pressure has been estimated by solving the Bernoulli equation (Wu et al., 2024a). Combined with the external pressure coefficient, the net wind pressure coefficient of the roof panel and its extreme value can be determined. For simplicity, Figure 4 and Figure 5 The AOA is 270° and the opening condition is =[1,1,1,0,0,0,1,1], the time history of the net wind pressure coefficient and the CDF of the extreme value of the net wind pressure coefficient of roof panels A and B, which also shows the corresponding results of the external wind pressure coefficient involved in these two roof panels, where Figure 4 (a) is the time history comparison diagram of the net wind pressure coefficient and the external wind pressure coefficient of roof panel A. Figure 4 (b) is a time-course comparison diagram of the net wind pressure coefficient and the external wind pressure coefficient of roof panel B; Figure 5 (a) is a comparison chart of the extreme net wind pressure coefficient and the extreme external wind pressure coefficient CDFs of roof panel A. Figure 5 (b) is a comparison of the extreme net wind pressure coefficient and the extreme external wind pressure coefficient CDFs of roof panel B. The comparison results show that the net wind pressure coefficient is greater than the external wind pressure coefficient, which makes the panel more susceptible to wind damage.

[0171] Substituting these variables, including the JCDF of the annual extreme wind speed in each direction, the probability of occurrence of different opening conditions under specific wind speeds in each direction, and the JCDF of the extreme net wind pressure coefficient under different wind speeds and opening conditions in each direction, into formula (17), the JCDF of the extreme wind pressure can be obtained. Figure 6 The JCDF contours of the extreme wind pressures for panels A and B, considering their directionality and multiple openings, are shown. Using Equation (15), it can be concluded that the extreme wind pressures for panels A and B at an MRI of 50 years are 1996 N and 1374 N, respectively. Clearly, the correlation between multiple extreme wind pressures affects their JCDFs.

[0172] 3.3 Analysis of wind-induced disasters

[0173] In this section, we will use the roof panel resistance as an input variable to calculate the probability of roof panel damage and thus derive the loss rate. Due to the directionality of wind speed and wind pressure, the damage rate of roof panels varies greatly in different structural directions. Therefore, this study will introduce the loss rate estimation for different structural directions. The structural direction is defined as the angle between the N wind direction and AOA = 0°, and is expressed as The failure probability of the roof panel in each structural direction can be calculated directly by the integral formula, i.e., formulas (24) and (26), or by analytical derivation, i.e., formulas (44)-(50) and (54).

[0174] Using formulas (31)-(38), it is easy to obtain the first eight order origin moments of the loss rate. The probability distribution of the loss rate can be reconstructed by MEM in Wang et al. (2021). In addition, they can also be obtained with samples generated based on MCS or through JTM (Wu et al., 2023b). Figure 7 Four cases were compared (i.e. =45°, 90°, 180°, and 270°). The results determined by the proposed MEM-based method are denoted by "Proposed MEM-based method," while those determined by MCS and JTM are denoted by "MCS method" and "JTM-based method," respectively. Note that these results account for the true correlation between variables (i.e., directional wind speed and wind pressure coefficient at the same directional wind speed). Overall, the proposed MEM-based method demonstrates satisfactory performance in estimating the probability distribution of loss rates. Clearly, the proposed MEM-based method better estimates the loss rates for critical MRIs (e.g., 10-year and 50-year) than the JTM-based method. Specifically, for the three structural orientations of 90°, 180°, and 270°, both the JTM-based method and the proposed MEM-based method perform well, with the proposed method outperforming the JTM-based method in the tail. In particular, for the 45° structural orientation, the JTM-based method clearly loses its effectiveness. This is primarily due to inaccurate JTM estimates of the damage rate parameters with skewnesses of 4.43 and 25.85, which are located near the boundary of the JTM application range. Satisfactorily, the overall trend of the damage rate CDF estimated by the proposed MEM-based method is highly consistent with that of the MC method, although there are some fluctuations in certain areas. These fluctuations are likely due to the failure of the obtained parameters to converge to their optimal solutions.

[0175] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not limiting. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solutions, which should all be included in the scope of the claims of the present invention.

Claims

1. A moment-based probabilistic analysis method for wind-induced vulnerability of low-rise buildings considering multiple hazard factors, characterized by: The following steps are involved: S1. Simplify the gradual damage of wind-induced disasters to the enclosure structures of low-rise buildings into a three-stage wind-induced disaster process; S2. A fully probabilistic framework for wind-induced vulnerability assessment of low-rise buildings under the combined effects of missiles and wind pressure, taking into account directionality and multiple openings, is proposed. This framework considers both directionality and multiple openings, including the calculation of the loss rate of windows / glass doors under different mean re-intensity intervals (MRIs) and the calculation of the loss rate of roof panels under different mean re-intensity intervals (MRIs). S3. Taking into account directionality and multiple openings, a closed-form formula for determining the first eight origin moments of the loss rate and an explicit expression for the failure probability of the envelope structure components are derived. Then, based on the origin moments, the maximum entropy model is used to obtain an estimated value of the loss rate of the envelope structure components under different MRIs, and an efficient analysis strategy for the wind-induced vulnerability of low-rise buildings is proposed.

2. The method for probabilistic analysis of wind-induced vulnerability of low-rise buildings according to claim 1, wherein: Step S1 is a three-stage wind disaster process: the first stage is a closed building with negative internal pressure, the second stage is a partially closed building with multiple openings on the wall and positive internal pressure, and in the third stage there are multiple openings on the roof panel, and the uplift force on the roof panel is smaller than the uplift force in the second stage.

3. The method for probabilistic analysis of wind-induced vulnerability of low-rise buildings according to claim 1, wherein: The specific method for calculating the loss rate of windows / glass doors under different mean refresh intervals (MRI) in step S2 is: S211. Determine the probability of WG damage caused by strong wind pressure; Determine the limit state function of the i-th WG damage caused by strong wind pressure in the k-th direction Expressed as: (2) in, and They represent the resistance and extreme wind pressure of the i-th WG in the k-th direction respectively; under the average wind speed in the k-th direction, the extreme wind pressure of the i-th WG is expressed as: (3) in, Indicates the i-th WG in the k-th direction The extreme value of the average wind speed in the kth direction Extreme wind pressure The cumulative distribution function (CDF) of is expressed as: (4) in, Indicates Conditional Conditional CDF of (i=1, 2, …, M); (i=1, 2, …, M) indicates a given Reaching wind pressure level Time , obtained from formula (3); according to , the probability of destruction and non-destruction of the i-th WG in the k-th direction and Calculated by the following equation: (5) (6) in, yes PDF; S212. Determine the probability of WG damage caused by wind-induced missiles The probability that the i-th WG is damaged by the wind-driven missiles in the k-th direction Expressed as: (7) in, is the area ratio between the WG and its surrounding walls; is the number of fragments of the dth (d=1, 2, …, D) type generated from the uth (u=1, 2, …, U) source building in the kth direction; ,in Indicates the area of the target building projected onto the ground; is the probability density function value of the udth type of debris in the kth direction falling on the center of the target building; represents the probability that the impact momentum of the ud-th type of debris in the k-th direction exceeds the impact resistance of the i-th WG; S213. Determine the loss rate of WG under different MRI conditions Indicates whether the i-th WG is damaged in the k-th direction, representing the damage situation The probability of occurrence is ,Right now , representing the undamaged situation The probability of occurrence is ,Right now ; Vector Used to represent the potential opening conditions in the kth direction, the total number of which is 2 M , among which indivual( 1, 2, …, 2 M )For opening working condition express; The loss rate of WG in the kth direction is expressed as: (8) Assuming that different WG failures are independent of each other, the probability of occurrence of each opening condition in the kth direction is Expressed as: (9) Based on formulas (8) and (9), the corresponding The CDF of can be expressed as: (10) in, , ( 1, 2, …, M-1) are variables equal to 0 or 1. The CDF of the loss rate of the WG considering the directionality is derived as: (11) in, Therefore (k=1, 2, …, K) is the condition Conditional CDF of yes JPDF; along with Get, R year MRI (denoted as ), calculated as: (12) in, yes The inverse function of .

4. The method for probabilistic analysis of wind-induced vulnerability of low-rise buildings according to claim 3, wherein: In step S2, the loss rate of the roof panel under different MRI conditions is calculated. The specific calculation method is: S221. Joint probability distribution of extreme wind pressure Similar to formula (3), the average wind speed of the jth roof panel in the kth direction is Extreme wind pressure (k=1, 2, …,K) can be determined as: (13) When considering the uncertainty of the extreme net wind pressure coefficient, the j-th roof panel with directionality and multiple openings is considered. The CDF of the extreme wind pressure (j=1,2, …, N) is expressed as: (14) in, Therefore and Conditional Conditional joint CDF (JCDF) for (k=1, 2, …, K); Therefore The JCDF of different opening conditions appears in each direction of the condition; yes JPDF for (k=1, 2, …, K); The wind pressure coefficients in different directions are usually considered to be independent of each other, so formula (14) becomes: (15) in, Therefore and (k=1, 2, …, K) is the condition CDF of Assuming that the wind speed and opening conditions in each direction are also independent of each other, formula (15) is further simplified to: (16) in, (k=1, 2, …, K) is the average wind speed in the kth direction Down Probability of occurrence; is the PDF of the annual extreme wind speed in the kth direction; Extending formula (15) to multiple roof panels and taking into account the directionality and multiple openings, the JCDF of multiple extreme wind pressures is obtained as follows: (17) in, is the kth direction and The conditional JCDF of multiple extreme net wind pressure coefficients is conditional; S222, Evaluation of wind-induced damage to roof panels under different MRI conditions The limit state function used to determine the damage of the jth roof panel caused by strong wind pressure in the kth direction for: (23) in, represents the resistance of the jth roof panel; represents the extreme wind pressure of the jth roof panel; considering the directionality and multiple openings, the jth roof panel is damaged and undamaged The probabilities are expressed as: (24) (25) in, is the PDF of the resistance of the jth roof panel; When there are N roof panels, the probability that all N roof panels are destroyed is and the probability that the jth (j = 1, 2, …, N-1) roof panel is damaged while the Nth roof panel is not damaged Respectively expressed as: (26) (27) in, It is the JPDF of resistance, ; Similarly, use represents the indicator variable, which represents whether the j-th roof panel is damaged after considering all directions. (or 0) indicates damage (or no damage). Taking into account the directionality and multiple openings, the loss rate of the roof panel is expressed as: (28) Corresponding The CDF of is expressed as: (29) in, , ( 1, 2, …, N-1) are variables equal to 0 or 1, and we get Afterwards, R year MRI (denoted as ) is calculated as: (30) in, yes The inverse function of .

5. The method for probabilistic analysis of wind-induced vulnerability of low-rise buildings according to claim 4, wherein: In step S221, the multiple integrals in equation (17) are solved using the MCS method, by get Multiple random samples (k=1, 2, …, K) are used to calculate and , The ensemble mean of .

6. In the probabilistic analysis method for wind-induced vulnerability of low-rise buildings as claimed in claim 5, there are two special cases for multiple extreme net wind pressure coefficients under a specific wind direction: independence and complete correlation. Therefore, formula (17) has two specific expressions. For the first case, when multiple extreme net wind pressure coefficients under the same wind direction are considered to be independent, the JCDF of the multiple extreme wind pressures is expressed as: (18) in, Therefore and Conditional CDF of (j=1, 2, …, N; k=1, 2, …, K); For the second case, when multiple extreme net wind pressure coefficients under the same wind speed are completely correlated, the JCDF of multiple extreme wind pressures is: (19) in, Therefore and CDF of the main extreme net wind pressure coefficient under the condition; When the wind speeds in different directions and the opening conditions are also independent of each other, we can conclude that: (20) For the case where both are considered independent, the JCDF is estimated as: (21) For the case where full correlation is considered, the JCDF is calculated as: (22) in, is the CDF of the dominant extreme wind pressure.

7. The method for probabilistic analysis of wind-induced vulnerability of low-rise buildings according to claim 6, wherein: The closed-form formula derivation method for the first eight-order origin moments in step S3 includes: Will Before ( =1, 2, ..., )-order origin moment is denoted as , 8th level It is derived from the following formula: (31) (32) (33) (34) (35) (36) (37) (38) Before calculation The failure probability is required for the origin moment , , , , , , , ( , , , , , , , =1, 2, …, N), these probabilities are obtained from Equations (24) and (26) according to different failure conditions.

8. The method for probabilistic analysis of wind-induced vulnerability of low-rise buildings according to claim 7, wherein: The method for constructing the display expression of the failure probability of the roof panel in step S3 includes: When the j-th roof panel considering the direction in formula (15) is obtained CDF, When the CDF of is known, based on formula (23), when assuming and irrelevant (e.g., Stewart et al., 2016), The CDF of is derived as: (39) based on , through transfer process theory, and the corresponding standard Gaussian variable It can be expressed as: (40) in, is the transfer function; is based on The inverse CDF of ; the inverse function corresponding to formula (40) is: (41) in, express The inverse function of .

9. The method for probabilistic analysis of wind-induced vulnerability of low-rise buildings according to claim 8, wherein: Wooden roof panels and There is no correlation between ( , =1, 2, …, N), and it is usually considered that the resistance of different roof panels is unrelated, that is, hour , hour , and Correlation coefficient The derivation is: (42) in, yes and The correlation coefficient between according to , and Correlation coefficient It is estimated by the following analytical formula: (43) in, is an analytical function, the correlation coefficient After determination, the failure probability , ,…, Calculated as: (44) (45) (46) (47) (48) (49) (50) in, express Joint PDF of dimensional standard Gaussian variables; ,…, is the covariance matrix, where , and , as follows: (51) (52) (53) in, ( =1, 2, …, N; =1, 2, …, N), other order covariance matrices are analogous to the rules; In formula (24) Expressed as: (54) It is concluded that equations (44)-(50) and (54) can be used to replace equations (24) and (26) to calculate , , , ,…, and More convenient.

10. A moment-based probabilistic analysis system for wind-induced vulnerability of low-rise buildings taking into account multiple hazard factors, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that: When the processor executes the program, the method for probabilistic analysis of wind-induced vulnerability of low-rise buildings as described in any one of claims 1 to 9 is implemented.

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