Bridging Elasticity and Structural Engineering With the Finite Volume Method

The theory of elasticity serves as a vital mathematical framework for studying deformations and stress distributions in elastic solid bodies and structural members subjected to external loads. However, the analysis of three-dimensional elasticity problems can be intricate and computationally demanding. To simplify the analysis, assumptions are made based on the geometry and boundary conditions, leading to quasi-three-dimensional models. This dissertation presents, for the first time, novel implementations of finite-volume based solutions for important classes of elasticity problems discussed in standard and advanced monographs on the theory of elasticity, namely: plane problems, torsion problems, and flexure problems of structural members.
For plane problems, which pertain to structural members subjected to loads acting solely in the plane of the structure, the dissertation formulates plane stress, plane strain, and generalized plane strain conditions within a parametric finite-volume framework. This framework is extended to analyze orthotropic and monoclinic materials. The developed finite-volume method (FVM) is verified using elasticity solutions for bending of rectangular cantilever beams under plane strain and plane stress assumptions and then applied to investigate deformation and attendant stress fields of multi-layered and heterogeneous beams with inclusions and porosities. Additionally, FVM is employed to assist in accurate shear characterization of advanced unidirectional composites in off-axis tension tests and Iosipescu shear tests.
This dissertation also addresses the study of prismatic bars with arbitrary cross sections bounded by a cylindrical surface and transverse planes with loadings applied solely on their end faces. Solutions to this class of problems play critical roles in structural engineering design of members of practical cross sections whose response to transverse loading is limited to bending, with twisting eliminated or minimized. By utilizing the principle of superposition, the complete equilibrium problem of an elastic bar was solved by decomposing the applied loading into four elementary loadings: extension, bending, torsion, and flexure. FVM-based approach was subsequently developed to analyze torsion of bars with curved boundaries and to assess the flexural response of beams with different cross sections. The accuracy and convergence of the FVM were validated through comparison with analytical solutions for cross sections with convex and concave boundaries.
Overall, FVM was demonstrated to successfully assess torsion-flexure behavior of various beam cross sections of practical interest in structural design. Upon comparison with three-dimensional finite element simulations of a series of cantilever beams, it also verified Saint Venant's principle often invoked in structural design by quantifying the extent to which end effects propagate into the beam.
The developed method fills the gap in the elasticity theory formulation and limited analytical solutions of complex problems, offering a powerful alternative to variational techniques. The findings obtained by the demonstrated accurate FVM analyses of a wide range of structural problems contribute to the design and development of safer and more efficient structures in various engineering fields. Moreover, the method’s transparent framework makes it readily accessible to the structural engineering community, democratizing the analysis and design process. A Graphical User Interface (GUI) developed for torsion problems of common structural engineering members has been employed successfully in the delivery of advanced mechanics of materials courses and made available to structural engineers in the industry and government laboratories.