Research on the Internal Force Solution for Statically Indeterminate Structures Under a Local Trapezoidal Load

The calculation of internal forces is a critical aspect in the design of statically indeterminate structures. Local trapezoidal loads, as a common loading configuration in practical engineering (e.g., earth pressure, uneven surcharge), make it essential to investigate how to compute the internal forces of statically indeterminate structures under such loads by using the displacement method. The key to displacement-based analysis lies in deriving the fixed-end moment formulas for local trapezoidal loads. Traditional methods, such as the force method, virtual beam method, or integral method, often involve complex computations. Therefore, this study aims to derive a general formula for fixed-end moments in statically indeterminate beams subjected to local trapezoidal loads by using the integral method, providing a more efficient and clear theoretical tool for engineering practice while addressing the limitations of existing educational and applied methodologies. The integral method is employed to derive fixed-end moment expressions for three types of statically indeterminate beams: (1) a beam fixed at both ends, (2) an an-end-fixed another-end-simple-support beam, and (3) a beam fixed at one end and sliding at the other. This approach eliminates the redundant equations of the traditional force method or the indirect transformations of the virtual beam method, directly linking boundary conditions through integral operations on load distributions, thereby significantly simplifying the solving process. Three representative numerical examples validate the correctness and universality of the derived formulas. The results demonstrate that the solutions obtained via the integral method align with software-calculated results, yet the proposed method yields analytical expressions for structural internal forces. Comparative analysis shows that the integral method surpasses traditional approaches (e.g., force method, virtual beam method) in terms of conceptual clarity and computational efficiency, making it particularly suitable for instructional demonstrations and rapid engineering calculations. The proposed integral method provides a systematic analytical framework for the internal force analysis of statically indeterminate structures under local trapezoidal loads, combining mathematical rigor with engineering practicality. The derived formulas can be directly applied to real-world designs, substantially reducing computational complexity. Moreover, this method offers a more intuitive theoretical case for structural mechanics education, enhancing students’ understanding of the mathematical–mechanical relationship between loads and internal forces. The research outcomes hold both theoretical significance and practical engineering value, establishing a solving paradigm for the displacement-based analysis of statically indeterminate structures under complex local trapezoidal loading conditions.