Analysis of shear lag in cylindrical tubes of arbitrary cross section

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A very general analysis is given of the phenomenon of shear lag in thin-walled cylindrical tubes, with single-cell cross sections of arbitrary shape, containing any number of concentrated longitudinal booms that carry direct stress only, and subjected to any longitudinal distribution of bending moment and torque. Two equations relating the distributions of direct and shearing stresses on the cross section are derived for the most general case where the tube is non-uniform because of an arbitrary longitudinal variation of wall thicknesses and boom areas. These equa­tions, which are remarkably simple in view of their generality, incor­porate all the requirements of equilibrium and compatibility and provide corrections to the stresses, curvature and twist calculated from the engineers’ theory of bending and torsion. They also govern the distri­bution of stresses arising from the application of self-equilibrating systems of tractions to the end cross sections. Exact solutions are ob­tained for the case of a uniform, but otherwise arbitrary, cross section under any polynomial distribution of bending moment and torque, and it is shown how conditions at the end cross sections can be satisfied with the aid of solutions of a simple eigenvalue problem. The equations are in a particularly ideal form for incorporating into a general purpose com­puter program for the automatic numerical solution of any problem of this type.