Keywords: nonuniform torsion, shear stresses, warping, bar, beam, twist, boundary element method, shear center, Wagner strain.

When arbitrary torsional boundary conditions are applied either at the edges or at any other interior point of the bar due to construction requirements, this bar under the action of general twisting loading leads to nonuniform torsion. Since thin-walled open sections have low torsional stiffness, the torsional deformations can be of such magnitudes that it is not adequate to treat the angles of cross-section rotation as small. When finite twist rotation angles are considered, the elastic nonuniform torsion problem becomes nonlinear. Moreover, this problem becomes much more complicated in the case that the cross section's centroid does not coincide with its shear center (asymmetric beams), leading to the formulation of a flexural-torsional coupled problem.

To the author's knowledge very little work, restricted to thin-walled beams, has been done on the aforementioned problem, while the numerical examples presented concern only uniform torsion of either mono- or doubly- symmetric cross sections.

In this paper the nonlinear elastic nonuniform torsion analysis of simply or multiply connected cylindrical bars of arbitrary cross section is presented. Three boundary value problems with respect to the variable along the beam axis angle of twist, to the primary and to the secondary warping functions are formulated. The first one, employing the analog equation method, a boundary element method (BEM) based method, yields a system of nonlinear equations from which the angle of twist is computed using an iterative process. The remaining two problems are solved employing a pure BEM based method. The proposed formulation procedure is based on the assumption of no local or lateral torsional buckling or distortion and includes the following essential features and novel aspects compared with previous ones:

1. Large deflections and rotations are taken into account that is the strain-displacement relationships contain higher order displacement terms.
2. For the first time in the literature, the present formulation is applicable to arbitrarily shaped thin or thick-walled cross sections occupying simple or multiple connected domains.
3. For the first time in the literature, both the linear and the nonlinear parts of the secondary warping shear stresses arising from the nonuniform torsion are evaluated as well as all the linear and nonlinear stress resultants including the nonlinear warping moment.
4. The presented formulation does not stand on the assumption of a thin-walled structure and therefore the cross section's torsional rigidity is evaluated exactly without using the so-called Saint -Venant's torsional constant.
5. The developed procedure retains most of the advantages of a BEM solution over a pure domain discretization method, although it requires domain discretization.

As observed, geometrical nonlinearity leads to stiffening of the structure and its better response against torsional loading, the nonlinear Wagner torque can reach significant values locally along the bar axis, bars of asymmetric cross section exhibit lateral displacement of their shear center axis under pure torsional loading, while the nonlinear warping moment can be neglected in most cases.

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