Shear and bending flexibility in closed-form moment solutions for continuous beams and bridge structures

Shear deformations can affect final member-end-moments for statically indeterminate continuous beams and frame structures, though for typical civil engineering structures their effect is small and moments can be based on flexural deformations only. When a member is deep relative to span length, however, shear deformations should be considered in the analysis. This can be included in the stiffness method and in a modified form of moment distribution where the carry-over factor is less than one-half due to the added flexibility from shear. In a prior paper the first author presented a new approach for solving statically indeterminate beams and bridge frames, with final end moments given in closed-form expressions. The advantages of this new approach are that no simultaneous equations are required as in the stiffness method, moments are not distributed back and forth as in moment distribution, and manual calculations may be used which give exact results for as many spans as desired. While only flexural deformations were considered in the original paper, this paper presents a closed-form approach that has been modified to include shear deformations. Final expressions are given for continuous beams and bridge frames, providing exact member-end-moments that match results from the stiffness method when shear deformations are included in the analysis.

► Shear flexibility has been added to the closed-form moment equations.
► The modified expressions are exact and require no simultaneous equations.
► For deep members, shear flexibility affects moments, shear forces and axial forces.