Low-rise building wind-induced vulnerability probability analysis method and system considering multiple disaster factors based on moment of inertia

Abstract

This invention relates to a moment-based probabilistic analysis method for the wind-induced vulnerability of low-rise buildings, considering multiple disaster-causing factors. First, it proposes a full-probability framework for assessing the wind-induced vulnerability of low-rise buildings under the combined effects of projectiles and wind pressure, considering directionality and multiple openings. This framework simultaneously considers directionality and multiple openings, including calculating the loss rates of windows / glass doors and roof panels under different MRI values. Based on this framework, closed-form formulas for determining the first eight orders of raw moments and explicit expressions for the failure probabilities of building envelope components are derived. Then, based on the raw moments, a maximum entropy model is used to obtain estimates of the loss rates of building envelope components under different MRI values, thus proposing a moment-based, efficient probabilistic analysis method for the wind-induced vulnerability of low-rise buildings. This wind disaster assessment method has the dual advantages of high accuracy and efficiency, helping to better meet the needs of the engineering and insurance industries, and numerical examples also verify the effectiveness of the assessment method.

Description

Technical Field

[0001] This invention belongs to the field of wind disaster assessment technology for low-rise building envelopes, and relates to a probability analysis method and system for wind-induced vulnerability of low-rise buildings considering multiple disaster-causing factors. In particular, it relates to a method and system for efficient and accurate assessment of the probability analysis of wind-induced vulnerability of low-rise buildings considering directionality and multiple openings under the combined action of projectiles and wind pressure based on moments. Background Technology

[0002] Low-rise buildings with wooden roofs are widely used in residential communities due to their low cost and large living space. Generally, these structures are very sensitive to wind and are easily damaged in strong winds (Ellingwood et al., 2004; Gurley and Masters 2011; Yang et al., 2018). Post-disaster investigations have found that the building envelope components, such as windows, glass doors, and roof panels, suffered severe damage under strong winds. This can be attributed to the stronger uplift force on the roof panels after the failure of windows / glass doors on the windward walls (Habte et al., 2017; He et al., 2018; Lan and Huang, 2022; Sharma and Richards, 2005). Notably, in addition to strong wind pressure, the impact of wind-blown debris is also a significant cause of window / 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 required.

[0003] Many researchers have used models built from wind disaster 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 to roof panels. The latter (i.e., reliability-based models) (Ellingwood et al., 2004; Stewart et al., 2016; Zeng et al., 2021) are likely more accurate for structures in areas with insufficient disaster data and are therefore more widely used. Vickery et al. (2006a; 2006b) developed a wind disaster and loss estimation model for low-rise buildings, which was used in the United States Multiple Hazards (HAZAUS-MH). Some researchers (Pinelli et al., 2011; Pita et al., 2012) have proposed public loss assessment models specifically designed to predict residential building losses in Florida. These models are primarily used for loss estimation during hurricane events in the United States, involving hurricane wind field and path simulation. Lee and Rosowsky (2005) analyzed the vulnerability of low-rise timber frame roof panels based on wind loads determined by specifications. Using wind tunnel data, some studies (Huang et al., 2015; He et al., 2015; Stewart et al., 2016) have effectively considered the uncertainty of pressure coefficients to assess damage to asphalt shingle roofs and metal roof cladding. These models mainly aim to estimate wind-induced losses of low-rise building envelopes in specific directions, without considering wind directionality. However, due to the significant directional effects of wind speed and aerodynamics, wind-induced losses of low-rise buildings vary significantly in different directions. Therefore, directionality should be considered in wind damage estimation. Ji et al. (2018) considered directionality in estimating wind-induced losses of steel roofs. Wu et al. (2023a) proposed a new wind damage estimation method that considers wind directionality, achieving high estimation efficiency. However, the extensive use of Monte Carlo simulations and multiple integrations makes these methods computationally inefficient. Wu et al. (2023b) proposed a full probability framework considering directionality and a moment-based method to more efficiently assess wind-induced losses to roof panels under different mean regression periods (MRI). However, this method does not account for multiple openings on the windward wall caused by projectiles, which may lead to an underestimation of wind damage. Summary of the Invention

[0004] In view of this, to address the problems of existing wind disaster assessment methods, such as overly risky estimations, low computational efficiency, and limited adaptability, which fail to meet the needs of the engineering and insurance industries, this invention provides a moment-based method and system for probabilistic analysis of wind-induced losses in low-rise buildings, considering multiple disaster-causing 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 values. This wind disaster assessment method can be considered an enhanced version of existing probabilistic wind disaster 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] To achieve the above objectives, the present invention provides the following technical solution:

[0006] A method for probabilistic analysis of wind-induced losses of low-rise buildings based on moments and considering multiple disaster-causing factors includes the following steps:

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

[0008] S2. A full probability framework for assessing the wind-induced vulnerability of low-rise buildings under the combined effects of projectiles and wind pressure, considering directionality and multiple openings, is proposed. This framework simultaneously considers both directionality and multiple openings, including calculating the loss rate of windows / glass doors under different mean re-renewal periods (MRI) and calculating the loss rate of roof panels under different mean re-renewal periods (MRI).

[0009] S3. Based on the consideration of directionality and multiple openings, the closed formula of the first eight orders of the original moments for determining the loss rate and the explicit expression of the failure probability of the building envelope components are derived. Then, based on the original moments, the maximum entropy model is used to obtain the estimated value of the loss rate of the building envelope components under different MRI, and then an efficient analysis strategy for the wind-induced vulnerability of low-rise buildings is proposed.

[0010] The moment-based probability analysis system for wind-induced vulnerability of low-rise buildings, which considers multiple disaster-causing factors, includes a memory, a processor, and a computer program stored in the memory and capable of running on the processor. When the processor executes the program, it implements the aforementioned probability analysis method for wind-induced vulnerability of low-rise buildings.

[0011] The beneficial effects of this invention are as follows:

[0012] This invention discloses a moment-based probabilistic analysis method for the wind-induced vulnerability of low-rise buildings, considering multiple disaster-causing factors. It establishes a full probability analysis framework for the wind-induced vulnerability of low-rise buildings under the combined effects of projectiles and wind pressure, taking into account directionality and multiple openings. Based on directionality and multiple openings, it derives explicit expressions for determining the failure probabilities of multiple building envelope components and closed-form formulas for calculating the first eight raw moments of the loss rate. Based on the raw moments, a maximum entropy model is used to obtain estimates of the loss rate of building envelope components under the mean regression period (MRI), proposing an efficient moment-based analysis strategy for the wind-induced vulnerability of low-rise buildings. Numerical examples of low-rise buildings based on wind tunnel tests are used to verify and demonstrate this method. Numerical case analysis results show that the estimated loss rate of building envelope components obtained by this method agrees well with the results of the MCS-based method. Furthermore, compared with the MCS-based method, it avoids a large number of Monte Carlo simulations, greatly improving computational efficiency. This estimation method features high accuracy and high computational efficiency, better meeting the needs of the engineering and insurance industries.

[0013] Other advantages, objectives, and features of the invention will be set forth in the following description, and in some respects will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0014] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0015] Figure 1 A simplified diagram of the three-stage wind-induced disaster process;

[0016] Figure 2 This is a diagram showing the building layout in a specific numerical example;

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

[0018] Figure 4 In the specific numerical example, AOA is 270° and the opening condition is... When the value is [1,1,1,0,0,0,1,1], the time histories of the net wind pressure coefficients of roof panels A and B are as follows: Figure 4 (a) is a 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 history comparison diagram of the net wind pressure coefficient and the external wind pressure coefficient of roof panel B;

[0019] Figure 5 In the specific numerical example, AOA is 270° and the opening condition is... When the value is [1,1,1,0,0,0,1,1], the extreme net wind pressure coefficients (CDFs) of roof panels A and B are given. 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 The extreme wind pressure JCDF between roof panels A and B in the specific numerical example;

[0021] Figure 7 The CDFs are the roof panel loss rates calculated using the proposed method in the specific numerical examples. Figure 7 (a) is Probability distribution of roof panel loss rate at 45° Figure 7 (b) is =90° Roof panel loss rate probability distribution diagram Figure 7 (c) is =180° Roof panel loss rate probability distribution diagram Figure 7 (d) is Probability distribution of roof panel loss rate at 270°. Detailed Implementation

[0022] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention.

[0023] A method for probabilistic analysis of wind-induced losses of low-rise buildings based on moments and considering multiple disaster-causing factors includes the following steps:

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

[0025] Post-disaster investigations revealed that windows, glass doors, and roof panels are vulnerable components of the building envelope of low-rise buildings. Although the actual wind disaster process is extremely complex, the main reasons can be summarized into two points: (1) the wind pressure on the building envelope components generally exhibits non-Gaussian characteristics and has large extreme values; (2) strong wind pressure or wind-induced projectile impacts usually lead to 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 panels 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 the damage. This simplified process includes three stages: stage 1: enclosed building, stage 2: partially enclosed building, and stage 3: roof panel loss. Assuming the building is an ideally enclosed structure, stage 1 has negative internal pressure, stage 2 is a partially enclosed building with multiple openings in the walls and positive internal pressure. In the third stage, there are multiple openings in the roof panel, which makes the upward pull force on the roof panel less than that in the second stage. Figure 1 The process is illustrated. It considers the changes in internal pressure caused by openings in the wall and their impact on the net wind pressure on the roof, and takes into account the worst-case scenario of wind damage to the roof, thus arriving at a relatively conservative estimate of wind disaster. More details of this process can be found in Wu et al. (2023a).

[0026] S2. Establish a full probability assessment framework for wind-induced vulnerability of low-rise buildings considering directionality and multiple openings under the combined effects of projectiles and wind pressure.

[0027] A full probability framework for assessing the wind-induced vulnerability of low-rise buildings under the combined effects of projectiles and wind pressure, considering directionality and multiple openings, is proposed. This framework simultaneously considers both directionality and multiple openings, including calculating the loss rate of windows / glass doors under different mean re-repair periods (MRI) and calculating the loss rate of roof panels under different mean re-repair periods (MRI).

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

[0029] Typically, the impact effect of strong wind pressure or wind-driven projectiles will cause the i-th (i 1, 2, …, M) Window / Glass Door Damage (hereinafter referred to as WG). Given the limited availability of relevant data and the lack of fundamental research in this field, the methodological system of this paper will temporarily exclude two potential correlations: (1) the correlation between the effects of strong wind pressure and wind-borne debris on building envelopes; and (2) the spatial correlation of the effects of strong wind pressure or wind-borne debris leading to correlated failures between different wall components. This simplification aims to maintain the feasibility of the analysis while reserving space for refining the theoretical framework with sufficient data. The i-th WG is the first ( 1, 2, …, The probability of destruction in the direction It can be represented as:

[0030] (1)

[0031] in, and This represents the probability that the i-th WG will be destroyed by strong wind pressure from the k-th direction and by wind-induced projectiles.

[0032] S211. Determine the probability that strong wind pressure will cause damage to WG;

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

[0034] (2)

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

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

[0037] (3)

[0038] in, kg / m 3 air density; Represents the i-th WG in the k-th direction. The extreme values ​​of, among which This represents the net wind pressure coefficient. Generally, it is recommended to use a transfer process method (e.g., Sadek and Simiu, 2002; Zhang et al., 2019) based on... get The probability distribution is equal to the external wind pressure coefficient. Subtract internal wind pressure coefficient ,Right now The internal pressure in stage 1 is usually caused by background leakage, which is negligible here. .

[0039] To more accurately estimate wind damage, it is usually necessary to consider the uncertainty related to the extreme wind pressure coefficient WG. The average wind speed in the k-th direction... Extreme wind pressure The cumulative distribution function (CDF) can be expressed as:

[0040] (4)

[0041] in, Indicates conditional Conditional CDF for (i=1, 2, …, M); (i=1, 2, …, M) represents a given When the wind pressure level is reached time The result is derived from formula (3).

[0042] according to The probability of destroying or not destroying the i-th WG in the k-th direction. and Calculate using the following equation

[0043] (5)

[0044] (6)

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

[0046] S212. Determine the probability that wind-induced projectiles will cause WG damage.

[0047] Strong winds can carry upstream debris such as branches, stones, and tiles, impacting building surfaces and causing damage to vulnerable components like windows / glass doors. These debris fragments are typically categorized as compact, sheet-like, and rod-like. Generally, it is difficult to establish a wind-induced projectile hazard model that considers the initial conditions, flight trajectory, and impact on buildings. Currently, the wind-induced projectile hazard model proposed by Lin and Vanmarcke (2010) is widely used for wind hazard assessment and analysis of engineering structures. Based on this model, the probability of a wind-induced projectile from the k-th direction causing damage to the i-th wind-induced projectile (WG) is calculated. It can be represented as:

[0048] (7)

[0049] in, It is the ratio of the area of ​​WG to the area of ​​its surrounding walls; It is the number of fragments of type d (d=1, 2, …, D) generated from the u-th (u=1, 2, …, U) source building in the k-th direction (hereinafter referred to as the ud-th type fragment); ,in This represents the area of ​​the target building projected onto the ground. Let be the probability density function value of the ud-th type fragment falling at the center of the target building in the k-th direction; This represents the probability that the impact momentum of the ud-th fragment in the k-th direction exceeds the impact resistance of the ith WG. Detailed estimation methods for the parameters in Equation (7) can be found in Ji et al. (2020). Clearly, determining these parameters typically requires building information and wind conditions.

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

[0051] This represents whether the i-th WG is destroyed in the k-th direction, and represents the destruction status. The probability of occurrence is ,Right now , representing an undamaged state The probability of occurrence is ,Right now ; vector This is used to represent the potential opening case in the k-th direction, and its total number is 2. M , of which indivual( 1, 2, …, 2 M Opening working condition express;

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

[0053] (8)

[0054] Assuming that different WG failures are independent (Ji et al., 2020), then the probability of occurrence of each opening condition in the k-th direction is... Represented as:

[0055] (9)

[0056] Based on formulas (8) and (9), the corresponding The CDF can be represented 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 directional WG can be derived as:

[0059] (11)

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

[0061] along with Obtained, under R-year MRI (recorded as) ), can be calculated as:

[0062] (12)

[0063] in, yes The inverse function of .

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

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

[0066] according to Changes in external wind pressure at openings will lead to changes in internal pressure within a building. Previous studies (e.g., Holmes 1979; Oh et al., 2007) have shown that many factors, including building volume, opening location, and number, influence internal pressure. To effectively account for the complexity and variability of opening conditions, Bernoulli's equation (e.g., Wu et al., 2024a) is commonly used to simulate internal pressure, offering satisfactory accuracy and efficiency. This study also employs Bernoulli's equation to simulate the internal pressure coefficient in the k-th direction. For more information, please refer to Wu et al. (2024a). Then, the net wind pressure coefficient in the k-th direction is... (j=1, 2, …, N) can be derived from Received, among which Let be the external wind pressure coefficient in the k-th direction and the j-th external wind pressure coefficient. The corresponding extreme net wind pressure coefficient is... can be It is obtained using the same method as WG.

[0067] Similar to formula (3), the average wind speed of the j-th roof panel in the k-th 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 taken into account. The CDF of 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 JCDF for different opening conditions in each direction; yes JPDF (k=1, 2, …, K).

[0072] Wind pressure coefficients in different directions are generally considered to be independent of each other (Tian and Chen, 2020), therefore equation (14) becomes:

[0073] (15)

[0074] in, Therefore and (k=1, 2, …, K) are the conditions CDF.

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

[0076] (16)

[0077] in, (k=1, 2, …, K) is the average wind speed in the k-th direction. Down The probability of occurrence; It is a PDF of the annual extreme wind speed in the k-th direction.

[0078] Furthermore, by extending formula (15) to multiple roof panels, taking into account directionality and multiple openings, the JCDF for multiple extreme wind pressures can be obtained as follows:

[0079] (17)

[0080] in, It is the kth direction downwards and Let JCDF be the conditional JCDF for multiple extreme net wind pressure coefficients. Clearly, the multiple integrals in equation (17) can be solved using the MCS method with satisfactory estimation efficiency. Firstly, it can be obtained through... get Multiple random samples (k=1, 2, …, K) are then used to calculate… and . The set average can be approximated as .

[0081] Furthermore, for multiple extreme net wind pressure coefficients under a specific wind direction, there are two special cases: independent and completely correlated. Therefore, formula (17) has two specific expressions. For the first case, when multiple extreme net wind pressure coefficients under the same wind speed are considered 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 scenario, when the net wind pressure coefficients of multiple extreme values ​​under the same directional wind speed are completely correlated, the JCDF of the multiple extreme wind pressures is:

[0085] (19)

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

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

[0088] (20)

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

[0090] (twenty one)

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

[0092] (twenty two)

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

[0094] S222 Assessment of wind-induced loss of roof panels under different MRI scans

[0095] In strong winds, whether the j-th roof panel fails depends on the resistance, uplift force, and self-weight. For buildings with lightweight timber frames, self-weight loads are typically neglected due to their lightweight nature; this study also excludes self-weight loads to obtain conservative results (Lee and Rosowsky, 2005). Therefore, the limit state function used to determine the failure of the j-th roof panel caused by strong wind pressure in the k-th direction is... It can be given by the following formula:

[0096] (twenty three)

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

[0098] Considering directionality and multiple openings, the j-th roof panel is damaged. and undamaged The probabilities can be expressed as follows:

[0099] (twenty four)

[0100] (25)

[0101] in, This is the PDF of the resistance of the j-th roof panel.

[0102] Given N roof panels, what is the probability that all N roof panels will fail? The probability that the j-th (j = 1, 2, ..., N-1) roof panel fails while the N-th roof panel remains intact. They can be represented as:

[0103] (26)

[0104] (27)

[0105] in, The JPDFs of resistance are generally considered to be mutually independent (Wu et al., 2023b), that is... Please note that there are very few shared connections between the log cabin panels, which leads to... and The correlation between them is very weak. Therefore, it is assumed that in formulas (26) and (27) and They are independent of each other.

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

[0107] (28)

[0108] Correspondingly The CDF can be represented as:

[0109] (29)

[0110] in, , ( (1, 2, …, N-1) are variables that are equal to 0 or 1. This yields… Later, MRI in year R (recorded as) It can be calculated as:

[0111] (30)

[0112] in, yes The inverse function of .

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

[0114] For simplicity, the following analysis will only cover... The calculation, the calculation and Same. As mentioned before, The probability distribution can be obtained from formula (29). However, there are multiple failure conditions in the calculation, and performing multiple integrations for each failure condition is very time-consuming. Therefore, this study proposes a moment-based method that uses the maximum entropy model (MEM) to obtain the probability distribution. and This model is widely used to obtain CDF estimates for non-Gaussian variables; for details, see (Wang et al., 2021). In the developed MEM-based wind disaster assessment method, the raw moments of the loss rate are required. Once the raw moments are known, MEM can be used to estimate... PDF, i.e. It's very easy to get from get and Therefore, the following section will introduce a method for estimating the raw moments of the loss rate, including deriving formulas for determining the first eight raw moments of the loss rate and the failure probability of the roof panel.

[0115] S31, Calculation Closed formula for the moment at the origin

[0116] Will The former ( =1, 2, ..., The first-order origin moment is denoted as To achieve a relative balance between computational efficiency and accuracy, this study employs... =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 the calculation The first-order origin moment requires failure probability. , , , , , , , ( , , , , , , , =1, 2, …, N), and these probabilities can be obtained from equations (24) and (26) according to different failure conditions. However, when N is large, these calculations involving integration are inefficient. The following sections aim to develop corresponding analytical formulas to obtain the failure probabilities of the roof panels more efficiently.

[0126] S32. Roof panel failure probability estimation

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

[0128] (39)

[0129] based on Through the theory of transmission processes, With the corresponding standard Gaussian variables It can be represented as:

[0130] (40)

[0131] in, It is a transfer function; Based on The inverse CDF. 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 them, that is ( , =1, 2, …, N). Furthermore, it is generally accepted that the resistance of different roof panels is unrelated (e.g., Stewart et al., 2016), i.e. hour , hour .therefore, and correlation coefficient This can be deduced as (Wuet al., 2023b):

[0135] (42)

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

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

[0138] (43)

[0139] in, These are analytic functions, and they take different forms depending on the context. These analytic functions are summarized in Table 2 of Wu et al. (2024b), and for simplicity, they are not listed here again. Correlation coefficient Once determined, the probability of failure , , ..., 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 standard Gaussian variables; , ..., Let be the covariance matrix. For the sake of simplicity, this study only gives . , and The details are as follows:

[0148] (51)

[0149] (52)

[0150] (53)

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

[0152] Note the formula (24) It can be represented as:

[0153] (54)

[0154] Obviously, equations (44)-(50) and (54) can be used to replace equations (24) and (26) for calculation. , , , , ..., and More convenient.

[0155] Specific numerical examples

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

[0157] To verify the accuracy of the full probability assessment framework for wind-induced hazards 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 As shown. The building is 19.2 meters long, 12.2 meters wide, and 3.7 meters high, with eight windows measuring 1.83 meters by 1.07 meters (width by height) symmetrically distributed on all walls. Figure 2 The Chinese character is represented as ~ The roof system consists of 80 wooden planks, secured to the frame with 6d standard nails. The wind pressure coefficient of the building envelope was obtained through wind tunnel testing (Ho et al., 2005), and the relevant data is publicly available. Figure 2 The blue "+" indicates pressure measurement points for recording wind pressure. These pressure measurement points can be used for various angles of attack (AOA), including AOA = 0°, 5°, 10°, …, 90° and AOA = 270°, 275°, 280°, …, 360°.

[0158] The above-described moment-based probabilistic analysis method for wind-induced losses of low-rise buildings, considering multiple hazard factors, demonstrates that the resistance of windows and roof panels, as well as the impact resistance of windows on windward walls, are necessary for wind damage loss estimation. The selection of these resistances references several studies (e.g., Lee and Rosowsky, 2005; Unikrishnan and Barbato, 2016; Wu et al., 2023b), and the nominal values ​​and coefficients of variation (COV) of their probability distribution models are summarized in Table 1. Considering the reduction in resistance due to 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. Hazard Analysis of Wind-Aided Projectiles

[0162] To account for window damage caused by wind-blown debris, a residential community layout was designed, such as... Figure 3 As shown. In this residential complex, the building within the circle is the target building, and the other buildings are the source buildings, whose structures are identical to the target building. Considering all these factors is extremely complex, as various damaged envelope components of upstream buildings (such as windows / glass doors, roof panels, and tiles) can become sources of debris in actual disasters. The purpose of this paper is to demonstrate and illustrate the method proposed above, thereby simplifying the sources of wind-borne debris, with only failed roof panels considered as debris type (sheet-like). Roof panel pressures are obtained using wind tunnel data through interpolation techniques based on appropriate orthogonal decomposition (POD), and peak pressures are estimated (e.g., Ji et al., 2018). The number of damaged roof panels is determined by comparing peak pressures and resistances through multiple simulations. Referring to Lin and Vanmarcke (2010), in the wind-induced projectile disaster model, Md = 4.08 kg, Ad = 0.143 m², and Tk = 1.58 s.

[0163] 3. Roof panel wind disaster analysis

[0164] 3.1. Annual extreme wind speed in a given direction

[0165] Based on the aforementioned moment-based probability analysis method for wind-induced losses of low-rise buildings considering multiple disaster-causing factors, the joint probability distribution of extreme wind speeds in each direction was calculated. This study also used the wind speed data from the numerical example in Wu et al. (2023b), which 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: N, NE, E, SE, S, SW, W, and NW. The Gumbel distribution proved to be an excellent model for the probability distribution of wind speed samples in each direction (Bezabeh et al., 2018). The joint probability distribution (JCDF) of extreme wind speeds in each direction was easily obtained using Gaussian copula. The determination process is described in Wu et al. (2023b).

[0166] 3.2 Extreme wind pressure

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

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

[0169]

[0170] For each opening condition The internal pressure has been estimated by solving Bernoulli's equation (Wu et al., 2024a). Combined with the external pressure coefficient, the net wind pressure coefficient of the roof panel and its extreme values ​​can be determined. For simplicity, Figure 4 and Figure 5 It demonstrates an AOA of 270° and an opening condition. When the value is [1,1,1,0,0,0,1,1], the time history of the net wind pressure coefficient and the CDF of the extreme values ​​of the net wind pressure coefficient for roof panels A and B are shown. The results also show the corresponding external wind pressure coefficients involved in these two roof panels. Figure 4 (a) is a 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 history 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), we can obtain the JCDF of the extreme wind pressure. Figure 6 The JCDF contour lines for extreme wind pressures, taking into account the directionality and multiple openings of panels A and B, are presented. Using Equation (15), the extreme wind pressures for panels A and B at MRI time of 50 years are 1996 N and 1374 N, respectively. Clearly, the correlation between multiple extreme wind pressures affects their JCDF.

[0172] 3.3 Analysis of Wind-Induced Hazards

[0173] In this section, roof panel resistance will be used as an input variable to calculate the probability of roof panel failure, thereby deriving the loss rate. Due to the directionality of wind speed and wind pressure, the roof panel damage rate varies significantly depending on the structural orientation. Therefore, this study will introduce loss rate estimation for different structural orientations, where the structural orientation is defined as the angle between the N-wind direction and AOA = 0°. The failure probability of the roof panel in each structural direction can be directly calculated by the integral formula, i.e., formulas (24) and (26), or it can be derived by the analytical formula, i.e., formulas (44)-(50) and (54).

[0174] Using formulas (31)-(38), the first eight raw moments of the loss rate can be easily obtained. For all 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 the help of samples generated based on MCS or by JTM (Wu et al., 2023b). Figure 7 Four cases were compared (i.e.) The probability distributions corresponding to the structural orientations (45°, 90°, 180°, 270°) are shown in the figures. The results determined by the proposed MEM-based method are denoted as "Proposed MEM based method," while the results determined by MCS and JTM are denoted as "MCS method" and "JTM-based method," respectively. Note that the true correlations between variables (i.e., directional wind speed and wind pressure coefficient at the same directional wind speed) have been considered in these results. Overall, the proposed MEM-based method demonstrates satisfactory performance in estimating the probability distribution of loss rates. Clearly, compared to the JTM-based method, the proposed MEM-based method better estimates the loss rates of critical MRIs (e.g., 10-year, 50-year). Specifically, for the three structural orientations of 90°, 180°, and 270°, both the JTM-based method and the proposed MEM-based method perform well, while the proposed method outperforms the JTM-based method at the tail end. Especially with a 45° structural orientation, the JTM-based method clearly loses its effectiveness, primarily due to the inaccurate estimation of damage rate parameters with skewnesses of 4.43 and 25.85 near the boundary of the JTM application range. Satisfactorily, the overall trend of the proposed MEM-based method in estimating the damage rate CDF is highly consistent with that of the MC method, although some fluctuations exist in certain regions. These fluctuations may be due to the obtained parameters failing to converge to their optimal solution.

[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 intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for probabilistic analysis of wind-induced vulnerability of low-rise buildings based on moments and considering multiple disaster-causing factors, characterized in that, Includes the following steps: S1. The gradual damage of wind-induced disasters to the building envelope of low-rise buildings is simplified into a three-stage wind-induced disaster process. In the three-stage wind-induced disaster process: the first stage is a closed building with negative internal pressure; the second stage is a partially closed building with multiple openings in the walls and positive internal pressure; and the third stage is a roof panel with multiple openings and the uplift force on the roof panel is less than the uplift force in the second stage. S2. A full probability framework for assessing the wind-induced vulnerability of low-rise buildings under the combined effects of projectiles and wind pressure, considering directionality and multiple openings, is proposed. This framework simultaneously considers both directionality and multiple openings, including calculating the loss rate of windows / glass doors under different mean re-repair MRI and calculating the loss rate of roof panels under different mean re-repair MRI. S3. Based on the consideration of directionality and multiple openings, the closed formula of the first eight orders of the original moment for determining the loss rate and the explicit expression of the failure probability of the enclosure structure components are derived. Then, based on the original moment, the maximum entropy model is used to obtain the estimated value of the loss rate of the enclosure structure components under different MRI, and then an efficient analysis strategy for the wind-induced vulnerability of low-rise buildings is proposed. Considering directionality and multiple openings, the closed-form formula for determining the first eight moments of the origin to establish the loss rate is derived as follows: Taking directionality and multiple openings into account, the loss rate of the roof panel is denoted as... ,Will The former , =1, 2, ..., The original moment is denoted as To achieve a relative balance between computational efficiency and accuracy, =8, using the formula to derive the 8th order .

2. The low-rise building wind vulnerability probability analysis method according to claim 1, wherein The specific method for calculating the loss rate of windows / glass doors under different mean re-exposure MRI in step S2 is as follows: S211. Determine the probability that strong wind pressure will cause damage to WG; Determine the limit state function for the damage caused by strong wind pressure in the k-th direction to the i-th WG. Represented as: (2) in, and Let represent the resistance and extreme wind pressure of the i-th WG in the k-th direction, respectively, where WG is the window / glass door; under the average wind speed in the k-th direction, the extreme wind pressure of the i-th WG is expressed as: (3) in, Represents the i-th WG in the k-th direction. The extreme values, Represents the net wind pressure coefficient; average wind speed in the k-th direction. Extreme wind pressure The cumulative distribution function (CDF) is expressed as: (4) in, Indicated by conditional Conditional CDF for i=1, 2, …, M; Let i = 1, 2, …, M represent the given... When the wind pressure level is reached time This can be derived from formula (3); According to , the probabilities of the kth direction i-th WG being broken and not broken and are calculated by the following equations: (5) (6) wherein is the PDF, PDF being the probability density function; S212. Determine the probability that wind-induced projectiles will cause WG damage. Probability of windborne missile caused by the kth direction to break the ith WG is represented as: (7) in, It is the ratio of the area of ​​WG to the area of ​​its surrounding walls; It represents the number of fragments of type d, d=1, 2, …, D generated from the u-th source building in the k-th direction; ,in This represents the area of ​​the target building projected onto the ground. Let be the probability density function value of the ud-th type fragment falling at the center of the target building in the k-th direction; This represents the probability that the impact momentum of the ud-th fragment in the k-th direction exceeds the impact resistance capability of the i-th WG; S213. Determine the loss rate of WG under different MRI scans. This represents whether the i-th WG is destroyed in the k-th direction, and represents the destruction status. The probability of occurrence is ,Right now , representing an undamaged state The probability of occurrence is ,Right now ; vector This is used to represent the potential opening case in the k-th direction, and its total number is 2. M , of which indivual, 1, 2, …, 2 M Open working condition express; The loss rate of WG in the k-th direction is expressed as: (8) Assuming that different WG breakages are independent of each other, the occurrence probability of each open working condition in the kth direction is represented as: (9) Based on equations (8) and (9), the corresponding CDF is represented as: (10) wherein, , , 1, 2, …, M-1 are variables equal to 0 or 1, the CDF of the loss rate of the directional WG is derived as: (11) wherein, is conditioned on , k = 1, 2, …, K the conditional CDF; is the JPDF; With Obtained, R-year MRI under , denoted as , calculated as: (12) wherein is the inverse function of 3. The method for probabilistic analysis of wind-induced vulnerability of low-rise buildings as described in claim 2, characterized in that, In step S2, the loss rate of the roof panel under different MRI scans is calculated. The specific calculation method is as follows: S221, Joint probability distribution of extreme wind pressure Similar to formula (3), the average wind speed of the j-th roof panel in the k-th direction is... Extreme wind pressure Let k = 1, 2, ..., K be determined as follows: (13) The extreme net wind pressure coefficient; considering the uncertainty of the extreme net wind pressure coefficient, the j-th roof panel with directionality and multiple openings is taken into account. The CDF of extreme wind pressures for j=1, 2, …, N is expressed as: (14) in, Therefore and conditional The conditional joint CDF for k=1, 2, …, K is called the JCDF. Therefore JCDF for different opening conditions in each direction; yes JPDF for k=1, 2, …, K; The wind pressure coefficients in different directions are generally considered to be independent of each other, so formula (14) becomes: (15) in, Therefore and k=1, 2, …, K are conditional CDF; Assuming that the wind speed and opening conditions in each direction are also independent, formula (15) is further simplified to: (16) in, k=1, 2, …, K is the average wind speed in the k-th direction. Down The probability of occurrence; It is a PDF of the annual extreme wind speed in the k-th direction; Extending formula (15) to multiple roof panels, and considering directionality and multiple openings, the JCDF for multiple extreme wind pressures is obtained as follows: (17) in, It is the kth direction downwards and The conditional JCDF is the net wind pressure coefficient with multiple extreme values ​​under certain conditions. S222 Assessment of wind-induced loss of roof panels under different MRI scans The limit state function used to determine the failure of the j-th roof panel caused by strong wind pressure in the k-th direction. for: (23) in, This represents the resistance of the j-th roof panel; This represents the extreme wind pressure of the j-th roof panel; considering directionality and multiple openings, the failure rate of the j-th roof panel is... and undamaged The probabilities are expressed as follows: (24) (25) in, This is the PDF of the resistance of the j-th roof panel; Given N roof panels, what is the probability that all N roof panels will fail? The probability that the j-th roof panel (j = 1, 2, …, N-1) fails while the N-th roof panel remains intact. They are represented as follows: (26) (27) in, It is the resistance JPDF, ; Similarly, using The index variable represents whether the j-th roof panel is damaged after considering all directions. 0 indicates damage or no damage. Considering directionality and multiple openings, the roof panel loss rate is expressed as: (28) Correspondingly The CDF is represented as: (29) in, , , Let 1, 2, …, N-1 be variables that are equal to 0 or 1, and obtain… Later, MRI in year R , recorded as The calculation is as follows: (30) in, yes The inverse function of .

4. The method for probabilistic analysis of wind-induced vulnerability of low-rise buildings as described in claim 3, characterized in that, In step S221, the multiple integrals in equation (17) are solved using the MCS method. get Given multiple random samples, k=1, 2, …, K, calculate using the samples. and , The set average can be approximated as .

5. The probability analysis method for wind-induced vulnerability of low-rise buildings as described in claim 4 has two special cases for multiple extreme net wind pressure coefficients under a specific wind direction: independent and completely correlated. Therefore, formula (17) has two specific expressions. For the first case, when it is considered that multiple extreme net wind pressure coefficients under the same wind speed are independent, the JCDF of multiple extreme wind pressures is expressed as: (18) in, Therefore and conditional CDF for j=1, 2, …, N; k=1, 2, …, K; For the second scenario, when the net wind pressure coefficients of multiple extreme values ​​under the same directional wind speed are completely correlated, the JCDF of the multiple extreme wind pressures is: (19) in, Therefore and CDF is the net wind pressure coefficient of the principal extreme values ​​under given conditions; 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 all variables are considered independent, the JCDF estimate is: (21) For the case where all correlations are considered, the JCDF is calculated as follows: (22) in, It is the dominant extreme wind pressure CDF.

6. The method for probabilistic analysis of wind-induced vulnerability of low-rise buildings as described in claim 5, characterized in that, Step S3, 8th order It is derived from the following formula: (31) (32) (33) (34) (35) (36) (37) (38) Before calculation The first-order origin moment requires failure probability. , , , , , , , , , , , , , , , =1, 2, …,N, these probabilities are obtained from equations (24) and (26) according to different failure conditions.

7. The method for probabilistic analysis of wind-induced vulnerability of low-rise buildings as described in claim 6, characterized in that, The method for constructing the explicit expression for the failure probability of the roof panel in step S3 includes: When the j-th roof panel considering directionality in formula (15) is obtained CDF, When the CDF is known, based on formula (23), when assuming and Irrelevant The CDF derivation is as follows: (39) based on Through the theory of transmission processes, With the corresponding standard Gaussian variables It can be represented as: (40) in, It is a transfer function; Based on The inverse CDF; the inverse function corresponding to formula (40) is: (41) in, express The inverse function of .

8. The method for probabilistic analysis of wind-induced vulnerability of low-rise buildings as described in claim 7, characterized in that, Wooden roof panels and There is no correlation between them, that is , , =1, 2, …, N, and it is generally believed that the resistance of different roof panels is unrelated, i.e. hour , hour , and correlation coefficient The derivation is as follows: (42) in, yes and The correlation coefficient between them; according to , and correlation coefficient The following analytical formula is used to estimate: (43) in, It is an analytical function, correlation coefficient Once determined, the probability of failure , , ..., The calculation is as follows: (44) (45) (46) (47) (48) (49) (50) in, express Joint PDF of standard Gaussian variables; , ..., Let be the covariance matrix, where , and The details are as follows: (51) (52) (53) in, , =1, 2, …, N; =1, 2, …, N, and other order covariance matrices follow the same pattern; In formula (24) Represented as: (54) Use equations (44)-(50) and (54) to replace equations (24) and (26) to calculate. , , , , ..., and .

9. A probability analysis system for wind-induced vulnerability of low-rise buildings based on moments and considering multiple disaster-causing 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, it implements the method for probabilistic analysis of wind-induced vulnerability of low-rise buildings as described in any one of claims 1 to 8.

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