## 5 Conclusions  
The fragility/vulnerability assessment of transmission tower structures subjected to winds is a multi-parametric issue. To this end, the paper proposes the surface expression (i.e., the surfaces \(\theta_{L}\) and \(\theta_{H}\)) for the wind-resistant capacity of the tower, based on the wind loading rule. The limit capacity of a certain tower is described by a function related to three external parameters, as well as by a function related to three wind load components, after which the corresponding limit sate functions of tower structures can be obtained. In order to acquire the capacity surfaces, the explicit algorithm-based non-linear static simulation is applied to the finite element model of the tower and a series of wind loading cases are involved. The simulation results yield the fitted/approximated capacity surfaces of an example tower by taking the transition point of capacity curves as the damage threshold. Applying the capacity surface \(\theta_{L}\) to the limit state function of the tower structure under winds, the capacity function term is constructed. Considering the structural fragility, the capacity function is high-dimensional and characterized by the stochasticity. For solving the high-dimensional probabilistic relation, a kriging-based adaptive modeling framework is established to approximate the capacity function. The constructed kriging model is expected to predict the points on the capacity surface \(\theta_{L}\), and these capacity points are characterized by stochasticity. Therefore, rather than a single surface, the kriging surrogate aims to approximate an assembly of surfaces. In the proposed modeling framework, the performance of the kriging surrogate is improved by adding support points. Three steps (i.e., the generation of support points, the optimization of the support point set, and the model validation) are involved to assure the effectiveness of the adaptive modeling. In the example study, the validation results (including the overall accuracy validation, the local error check and the practical issue-based validation) demonstrate the feasibility of the framework with all of the accuracy/error measures approaching to convergence/stationarity after a sufficient number of loops. Figure 16: Failure bounds of each strain section. (**a**) the lower bound of failure probabilities; (**b**) the upper bound of failure probabilities. By applying the established kriging surrogate of the capacity surface, a group of structural fragility curves for transmission towers are obtained, in which geo-spatial information of the line and wind (i.e., the line direction, the line span and the wind direction) is considered simultaneously. Depending on the fragility model, it is capable to do the fragility assessment on transmission towers and line under winds. Considering the safety of tower structures under typhoons, the annual failure probability of each tower in a practical transmission line is calculated by combining the capacity-surface-based fragility model with the joint probability model of the annual extreme wind speed and wind direction of typhoon. By the failure risk assessment on each strain section of the line, some suggestions are made for evaluating and improving the safety of the transmission line. Taking the line direction and span, the wind direction, and intensity into account, this paper explores an effective approach to establish and solve the capacity function for transmission towers under winds, and then assess the structural fragility of tower structures and transmission line. To do the fragility analysis more precisely, much more factors are needed to be involved (e.g., the wind stochasticity, the variance of terrain, and the service time of the transmission line). Correspondingly, a higher dimensional model for the wind-resistant capacity of the tower is required. Conceptualization, Y.C. ; methodology, Y.C. ; software, Y.C. ; validation, J.W. ; formal analysis, J.W. ; investigation, Y.C. ; resources, Y.C. ; data curation, Y.C.