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Risk assessment and life cycle cost analysis of electric power systems facilitate analysis and efficient management of compound risks from wind hazards and asset deteriorations. Fragility functions are key components for these analyses as they provide the probability of failure of poles given the hazard intensity. Despite a number of efforts that analyzed the wind fragility of utility wood poles, impacts of key design variables on the likelihood of failure of poles have not been yet characterized. This paper, for the first time, provides a set of multi-dimensional fragility models that are functions of key factors including class, age, and height of poles, number and diameter of conductors, span length, and wind speed and direction. Unlike existing generic pole fragility models, this new class of fragility functions is able to accurately represent various configurations of power distribution systems. Therefore, it can reliably support decisions for installation of new or replacement of existing damaged or decayed poles. The generated fragility models are also used to investigate impacts of design variables. For example, results indicate that when height is considered as a covariate for the fragility function, the likelihood of failure of wood poles for a given height increases with class number. However, if height is treated as an uncertain variable, and therefore, excluded as a covariate from the fragility model, lower classes of poles may have higher failure probability as they are often used for higher clearance limits.