The work considers the issues of the deformed state of the span beams of the bridge. The objective of the study is to develop a new mathematical model of the main beam of overhead cranes, which enables the investigation of its stress-strain behavior under the combined action of vertical moving loads, static loads from the weight of the beam itself, and axial eccentric loads. The proposed new mathematical model is based on the general theory of stability of elastic systems, taking into account the actual conditions of the crane bridge design. The obtained equations for the deflection curve of this beam allowed for the investigation and analysis of one of the main criteria for the performance of lifting machines - static stiffness and static strength, depending on the ratio of longitudinal and transverse forces acting on the beam. The results obtained in this work can be further used in the design and modernization of cranes to increase their lifting capacity, extend their service life without dismantling, as well as improve existing engineering calculation methods both at the design stage and under real operating conditions.