Saint Venant's Torsion by the Finite-Volume Method

Torsion introduces additional shearing in the structural element’s cross section, which may potentially produce failure in structures or damage their serviceability. Thus, the torsional deformation mode, characterized by the twisting of a structural element about its axis, plays an important role in structural engineering design. In this thesis, a new approach to Saint Venant’s torsion problems has been developed for the first time based on the finite-volume method. The approach employs the displacement formulation expressed in terms of the warping function subject to Neumann-type boundary condition to ensure traction-free lateral surface. The finite-difference method was also implemented as a reference for comparison and validation of the developed finite-volume method. Homogenous isotropic rectangular cross sections employed in structural engineering problems were analyzed by the finite-volume and finite-difference methods and validated against exact elasticity solutions. The convergence and accuracy of the finite-volume method relative to elasticity solutions were also demonstrated for composite cross sections made up of two symmetrically joined rectangular regions filled with different materials. Three typical homogenous cross sections employed in structural engineering were then analyzed in order to assess the accuracy of the membrane analogy widely used in the design. The finite-volume method’s strength lies in its superior ability to handle heterogeneous material cross sections, by inherently satisfying traction and displacement continuity conditions at the interfaces separating different materials, whereas the finite-difference method requires more refined grids to yield converged results. This strength was further demonstrated for composite cross sections with isotropic and orthotropic materials in the form of discontinuous and continuous reinforcement of concrete T-beam and box-beam cross sections. This thesis lays the foundation for the implementation of the finite- volume method in a large range of applications involving the design of composite structural elements with complex heterogeneous microstructures.