The failure of urban street trees caused by strong winds and has several adverse effects on urban functions and public safety. This study developed a wind fragility model based on the mechanical analysis of urban street trees. The uncertainty of the important parameters involved in this model was quantified for species of interest. Specifically, the vine copula function was used to estimate the joint probability distribution of the geometric parameters. Furthermore, the tree fragility curves were obtained and then validated by the historical measured date. The proposed model may help in effectively identifying highrisk streets and regions.
The breakage and falling of street trees caused by extreme winds may lead to direct economic losses and indirect impacts, such as the failure of structures, lifeline facilities, and traffic systems, which severely threaten the lives and properties of residents.
Existing studies on wind damage to trees can be roughly categorized into mechanistic and statistical approaches. The statistical approach directly estimates the probability of damage and the factors influencing tree failures from historical data using various regression and statistical techniques^{[1,2]}. However, because the date used for statistical analysis is typically locally specific, it has not been confirmed whether the developed statistical model can be generalized^{[3,4]}. Moreover, statistical models face difficulties in elucidating the actual mechanism of wind effects on trees.
In contrast, the mechanistic approach is predominant in studies on windinduced tree failure. As early as 1881, Greenhill investigated the stability of trees using a bottomfixed tapered rod^{[5]}. Subsequently, researches on the mechanistic tree model developed toward two directions. One direction tends to develop increasingly more refined and sophisticated tree models^{[6,7]}. The finite element model considering precise tree geometry and windtree interaction is particularly popular^{[814]}. The other direction is more practical and applicationoriented. It tends to use models with relatively simple geometry, and focuses on the specification and quantification of the key model parameters. Specifically, HWIND^{[15]}, GALES^{[16]}, and FOREOLE^{[17]} are the most widely investigated models. The three abovementioned modeling approaches simplify the tree as a tapered rod and adopt quasistatic analysis^{[18]}. The widely acknowledged Hazus model for multihazard loss estimation^{[19]} involves a module for tree blowdown, whereby an individual tree is modeled as a singledegreeoffreedom (SDOF) oscillator. However, the accuracy of this model is still unsatisfactory.
To evaluate the wind risk, most mechanistic models, such as HWIND, GALES, and FOREOLE, resort to calculating the critical wind speed (CWS) required to cause the damage to an "average tree"^{[18,20]}. In contrast, the Hazus tree blowdown model estimates tree damage using the fragility curve, that is, by calculating the conditional failure probability corresponding to different wind speeds. Owing to the inherent randomness of tree morphology, the environment, and tree failure occurrence, this study adopted the fragility scheme for risk assessment.
However, almost all the abovementioned studies have focused either on forest trees or simply on individual trees. Despite the severe and potentially catastrophic impact of urban tree failure^{[21,22]}, and the significant differences between the wind environment and tree features of urban areas and forests^{[23]}, few studies have established quantitative wind risk assessment models for urban trees^{[24]}. Studies focusing on trees in urban areas have practical significance, while the computational efficiency of the singletree model is important for facilitating subsequent cityscale analysis.
This study developed a wind risk assessment model for trees in urban streets. To predict windinduced failure, a mechanistic model based on the HWIND and GALES models is proposed. Further, focusing on the quantification of the involved uncertainties, the probability distributions of significant parameters were investigated in detail. Specifically, the joint probability distributions of the geometric parameters were estimated using the vine copula function, based on the data from an urban tree database. Finally, the tree fragility curves were obtained and validated.
A mechanistic tree model for the deterministic analysis of a single tree subjected to wind loading is proposed. The proposed model combines the HWIND and GALES models, and involves the simplification of trees into a cantilever corresponding to the trunk, as shown in
Diagram of tree model.
The basic parameters required to determine the proposed structural model are geometric and material parameters.
The three most fundamental geometric parameters of trees are the diameter at breast height (
Except for the three fundamental parameters, the taper equation of the trunk is essential for determining the element diameters. The proposed model adopts the widely used Max and Burkhart equation, which is expressed as follows^{[25]}:
where
The crown shape is an important geometric factor for tree modeling because it directly affects the estimated wind loads on trees. The following relationship is used to fit the crown profile^{[26]}:
where
The material parameters include the stem density
The selfweight of trees is calculated using elements. For elements in the crown area, the weight of the element is the combination of the stem and crown weight. The weight of the stem can be determined from the stem density
The mean wind speed at the top of the tree
where
Because quasistatic analysis is adopted for the proposed tree model, the effect of the fluctuating wind is considered by introducing a gust factor, as follows:
where
The wind load on each element is represented by a point load concentrated at the center of the element, as follows:
where
where
The overall and local failure of trees are considered. The overall failure includes the stem breakage mode and uprooting (or overturning) mode, while local failure refers to the branch breakage mode, which is typically more likely to occur compared with overall failure, but has a negative impact on the urban road environment as well.
Typical section diameter and stress of broadleaf species. (A) Section diameter; (B) Section stress.
Description of failure modes



Overall Failure  Uprooting  Base bending moment ( 
Stem breakage  Maximum section stress in the trunk region ( 

Local Failure  Branch breakage  Maximum section stress in the branch region ( 
The failure criterion for uprooting is expressed as follows:
In this equation, the critical base moment
where
The bending moment of each node is obtained as follows:
where
Notably, the additional moment in the HWIND model, which accounts for the pdelta effect caused by the selfweight and windinduced horizontal displacement, is not considered herein because the proposed model only considers the linearelastic response of trees. Consequently, the additional moment is relatively small compared with the moment generated directly by the wind loads.
Comparison of additional moment and direct moment.
The failure criterion for stem breakage is expressed as follows:
where
where
where
The failure criterion for branch fracture is expressed as follows:
where
In this section, the stochastic parameters involved in the proposed mechanistic model are quantified. Specifically, the joint distributions of the geometric parameters are fitted using data from the USDA urban tree database^{[27]} via copula functions. However, owing to the lack of data, the probability distributions of material parameters are given empirically with mean values obtained from the literature. The distribution of the wind gust factor is identified from the time histories of the fluctuating winds generated using the Davenport spectrum^{[30]}.
Because the geometric and material parameters of trees are speciesdependent, to facilitate the model validation discussed in the next section, typical broadleaf trees and conifers were considered as the target species in this study. Notably, the same methodology can be applied to the analysis of other specific species when the corresponding information is provided.
The basic geometric parameters, that is, the diameters at breast height (
The data used for regression were obtained from the USDA urban tree database^{[26]}. For each species, the regression parameters and the distribution of random variables were fitted. The model employed to fit the data points is expressed as follows:
where
The distributions of random variables
Part of data of geometric random variables
The linear polynomial function is employed for the regression of diameter at breast height (
Regression results for
Regression parameters for different species and geometric parameters (for linear function,




Broadleaf 



Conifer 



Before constructing the joint distribution of the random variable using copula theory, the marginal distributions of the variables were determined. In this study, five distribution types, namely, the normal distribution, lognormal distribution, Gamma distribution, Weibull distribution, and Burr type XII distribution, were considered and fitted by the maximum likelihood estimation.
Fitting results for assumed marginal distributions of geometric random variables of broadleaf trees.
The D value in the KolmogorovSmirnov (KS) test was used to determine the bestfitted distribution, as follows^{[31]}:
where
Bestfitted distribution and parameters of geometric random variables












Broadleaf  Burr 

Burr 

Burr 

Conifer  Burr 

Gamma 

Weibull 

The joint distributions of the geometric random variables (
Correlation coefficients for each random variable pair




Pearson  [ 
0.659  0.705 
[ 
0.173  0.088  
[ 
0.144  0.100  
Spearman  [ 
0.711  0.729 
[ 
0.182  0.087  
[ 
0.135  0.152  
Kendall  [ 
0.525  0.545 
[ 
0.124  0.060  
[ 
0.091  0.103 
Herein, five copula function types were considered: the AliMikhailHaq (AMH) copula, Frank copula, Clayton copula, Gaussian copula, and t copula. According to the Akaike information criterion (AIC), the bestfitted copula has the minimum AIC value^{[33]}. For each random variable pair, the bestfitted copula functions and their parameters are presented in
Comparison of samples generated by copula function and original data.
Best fitted copula function and their parameters












Broadleaf  t 

t 

t 

Conifer  Frank 

t 

Frank 

In the proposed model, the random material and strength parameters of trees include the stem density
where
The construction of accurate probabilistic distributions of the model parameters is always desirable. However, because a large data set of material parameters is not yet available, their distributions must be assumed based on experience. In the proposed model, the lognormal distribution is adopted for all random material and strength variables. The distribution parameters are determined by moment estimation; therefore, the mean value
The USDA Wood Handbook^{[34]} provides the specific gravity (
where
The mean values of
Mean values of





Broadleaf  86  0.517  923.9  5.57 
Conifer  105  0.401  802.2  4.34 
Owing to the lack of supporting materials, the crowntostem weight ratio
The regression coefficient
The conclusions regarding the random variables for the stem density, modulus of rupture, crowntostem weight ratio, and regression coefficient of the critical base moment, that is,
Random variables of material and strength properties




Lognormal 




Lognormal 




Lognormal 


Lognormal 

In this study, the gust factor was calculated by the ratio of the 1second gust to the 10min mean wind speed. The fluctuating wind speed can be considered as a Gaussian stationary process. For an initial variate satisfying the Gaussian distribution
where F
As can be seen, if
For each
The probabilistic mechanical model was validated in the same manner using the same survey data as the Hazus model.
In the technical manual of HazusMH 2.1, the blowdown results for 1158 trees (628 conifer and 530 deciduous trees) in eight residential subdivisions in eastern North Carolina caused by Hurricane Isabel in 2003, and the estimated peak gust wind speed of each subdivision, are provided^{[19]}. Trees are divided into conifers and deciduous trees by species, and into four classes by height, amounting to a total of eight classes. Because the sample size at each surveyed site is not sufficient to allow a reasonable estimation of the failure property if considered separately, the Hazus manual proposes a weighted average scheme to make use of these measured data^{[19]}. For each tree class at all surveyed sites, the weighted average blowdown probability for trees, and the corresponding weighted wind speed, were calculated as indicated by the star symbol in
Model validation results.
where U
By employing the proposed model, the failure probability
The survey data and estimation results for eight tree classes are shown in
Another phenomenon observed in the estimation is that the dominant failure mode is uprooting at the weighted wind speed. Actually, there is certain regularity between the dominant failure mode estimated by the proposed model and the wind speed. When the wind speed is low, the dominant failure mode is stem fracture, while the failure probability in this stage is typically too low to be heeded. When the wind speed increases, the trees are more likely to fail at the root.
As a further exploration, the significance of the correlation of random model parameters, that is, the necessity of applying the vine copula functions, was investigated. As shown in
Comparison of fragility curves of typical deciduous tree.
This paper proposes a wind risk assessment model for trees in urban streets. Specifically, based on a mechanistic tree model with elaborated considerations of the model parameter uncertainties, a probabilistic model for calculating the fragility of urban street trees was established. By using a sufficient amount of available data, the joint probability distributions of the fundamental geometric parameters of trees were obtained based on the vine copula theory. Moreover, the wind fragility curves of typical broadleaf trees and conifers were obtained. The comparison of the results obtained by the proposed model to survey data confirmed the validity of the proposed model.
The significance of accurate probability modeling in risk assessment was emphasized. In addition to the marginal probability distribution of each parameter, the correlation and dependence of parameters are also important. Copula functions provide a feasible approach toward accurately constructing the joint distributions of random parameters using parametric and explicit expressions.
The proposed mechanistic model is fairly efficient and is expected to be applied to wind risk assessment at the city scale. However, owing to the lack of data, there are still many deficiencies in the parameter quantification and model validation of the current model. Issues to be further improved include the following: (1) accurate modeling of the distribution of material and strength parameters with the support of more experimental data; (2) considering the influence of root characteristics and rootsoil interaction; and (3) comprehensive validation of proposed model with the support of additional survey data and refined numerical analysis.
Performed data gathering and analysis work: Luo Y
Designed the research work and provided guidance: Ai X
Not applicable.
None.
Both authors declared that there are no conflicts of interest.
Not applicable.
Not applicable.
© The Author(s) 2022.