抄録
In this report the torsion problems of thin cylindrical shells, of "open" or "closed" cross-section, with axial constraints are investigated.
St. Venant torsion theory, when applied to thin open sections, shows that the torsional strength and rigidity are very small. Even such an open section, however, is capable of transmitting an appreciable torque, when one end is fixed and during twist the warping of the cross-section is constrained axially. This can be explained as so-called bending-torsional action (or torsion-bending action).
Similary, in the case of thin closed sections, it is also clear that the axial direct stress, with additional shearing stress and direct stress perpendicular to the axial one, is introduced in addition to the stress given by the Bredt-Batho formula, when one end of the cylindrical shell is constrained axially.
From this point of view, the theory can be developed in two cases separately:-In one case the deformation of the shape of the cross-section is prevented by means of the bulkheads which have no stiffness out of their planes and are situated at frequent intervals along the axis of the cylindrical shell. In the other, it is assumed that the shape of the cross-section is deformable in its plane, neglecting the flexural rigidity of the wall and any existing transverse frames.
Comparing these two cases, it can be understood in a monocoque construction that the transverse frames, which prevent the deformation of the shape of the cross-section, have an effect to make rapidly decay the local perturbations of stresses due to axial constraints and also some effect for increasing torsional stiffness.
In Part I the general fundamental equations of equilibrium and the relation between the stresses and strains in thin shells of constant cross-section are discussed. In Part II the torsion theory is treated, assuming the deformation of the shape of the cross-section is perfectly prevented in its plane. The solution is given in Eqs. (23), (28) and (29), the simplified forms of whichare given in Eqs. (31). A method of determining the centre of twist is explained in §5. The relation between the angle of twist and the twisting moment is given by Eq. (41), and approximately by Eq. (55) when the distribution of the twisting moment and consequently the change of twist are monotonic. The "bending-torsional constant" CBT is given by Eq. (54) which is applicable for both open and closed sections. Eq. (55) differs a little from the well-known equation (55') which H. Wagner and W. Pretschner have introduced for open profiles(2). Rigorously speaking, however, our equation (55) is more correct.
In §7 end conditions are explained. Eqs. (60) and (62) correspond to the fixed and free end conditions, and Eq. (63) corresponds to the condition at a cross-section, where the twisting moment distribution abruptly changes.
Some examples are treated as follows:-The case in which one end is fixed and a concentrated twisting moment is applied at the other end, the case in which one end is fixed and an uniform twisting moment is distributed, and the case in which a concentrated twisting moment is applied at a certain cross-section between the fixed and the free ends, are given in §8, §9 and §10 respectively.
In Part III, a numerical example of a rectangular tube is calculated and the results almost coincide with those which D. William has already obtained by a different method. Another numerical example is also treated about a thin-walled elliptical tube. In Part IV the torsion problem of thin shells, the cross-sectional form of which is assumed deformable in its plane, is investigated, especially in the case where the shells have only bulkheads at the free and fixed ends and a twisting moment is applied at the free end.
