The safety risk analysis of urban elevated bridge construction is an important management method to reduce the loss of safety accidents, and it has significant scientific research value and engineering application value. Therefore, this study proposes a novel analysis method to address these challenges. Firstly, this paper constructs a Work Breakdown Structure (WBS)–Risk Breakdown Structure (RBS) matrix for the safety risk of urban elevated bridge construction in order to achieve a comprehensive and complete identification of the indicator system. Then, a combination of static weights and dynamic weights calculation methods is developed. The static weights are obtained using the analytic hierarchy process, while the dynamic weights are obtained based on the relationship between the dynamic scores of construction safety risk indicators in different construction stages and the preset evaluation levels. Finally, a case study of the Longlingshan elevated bridge project in Wuhan, China, is conducted to validate the feasibility of the proposed model and its potential application in projects. The case analysis for the first time reveals that with the progress of construction, the weights of each indicator continuously change, and the secondary indicators related to environmental factors, such as extreme high-temperature weather, undergo the greatest changes. A comparison of different dynamic weight calculation methods is conducted to highlight the advancement of the proposed model. The research findings of this paper will provide new insights and guidance for improving the construction safety of urban elevated bridge projects.