Torsion of Cylinders with Multiple Reinforcement

An exact solution is presented for the Saint-Venant torsion of a circular cylinder reinforced by multiple rings of circular rods. The number of equally spaced rods in each ring is the same, and these are arranged in such a way that the corresponding rods in each ring are located on the same radial line. The problem simulates that encountered in matrix rods reinforced by longitudinal fibers, and also in corresponding problems of reinforced concrete. Formulas are obtained for the boundary stress distributions and torsional rigidities. Stress function formulations are made for the torsion of cylinders having multiply connected composite sections. The stress function for each rod is expressed in terms of different polar coordinates, and use is made both of periodicity and symmetry. The degenerate case of a circular cylinder having multiple rings of holes is also considered. Several numerical examples are worked out, and the results are presented in tabular and graphical form.