Closed-form shear flow solution for box–girder bridges under torsion

To provide desired stiffness and strength in torsion, bridge superstructures are often constructed with a cross-section consisting of multiple cells which have thin walls relative to their overall dimensions and resist Saint–Venant torsion through shear flow (force per unit length) that develops around the cell walls. For a single thin-walled cell subject to torsion, shear flow is constant along each of its walls while shear stresses vary around the section based upon changes in wall thickness. When the cross-section contains multiple cells they all contribute resistance to applied torsion and for elastic continuity each cell must twist the same amount. With these considerations, equilibrium and compatibility conditions allow simultaneous equations to be formed and solved to determine the shear flow for each cell. A second approach is a relaxation method that distributes incremental shear flows back and forth between cells, reducing errors with each distribution cycle, until the final shear flows for all cells approximate the correct values. A major advantage to this method is that it does not require setting up and solving simultaneous equations, favoring situations where hand calculation is desired. In this paper, a closed-form approach is introduced to determine, exactly, both the torsional constant and all shear flows for multi-cell cross-sections under torsion; no simultaneous equations are required and there is no need to distribute shear flows back and forth between cells. Simple closed-form equations are derived which give shear flows for cross-sections with any number of cells of arbitrary shape.