Fundamental solutions for beams, plates, and shells under thermomechanical actions
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As the engineering profession moves from prescriptive or “deemed-to-satisfy” approaches towards design methodologies based on quantification of performance, sophisticated modelling tools are increasingly needed, especially when complex combinations of demand and capacity are encountered. Recourse is invariably made to advanced computational tools to provide high fidelity solutions to large and complex problems, such as the response of structural systems or components to thermomechanical actions. Software packages based on the finite element method are most commonly used for such analyses. There are some essential prerequisites to effective use of advanced computational software for complex nonlinear problems, which are often ignored, particularly in professional practice. These include a thorough understanding of the underlying mechanics of the problem under consideration; a good appreciation of the approximation methods for modelling the problem properly (e.g. the choice between elements, continuum or structural, low or high order interpolation, degree of mesh refinement necessary and so on); and perhaps most importantly ensuring that the software is reliable and is able to reproduce established fundamental solutions to an acceptable degree of accuracy. This thesis attempts to address most of these issues but focusses primarily on the last mentioned prerequisite and provides a range of novel and unprecedented fundamental solutions for beams, plates, and shallow shells subject to moderate or extreme thermomechanical loads such as those resulting from a fire. Geometric and material nonlinearities are included in the proposed formulations along with the most common idealised boundary conditions. Thermally induced deformations generate large displacements and require the solutions to account for geometric nonlinearity, while material nonlinearity arises from the degradation of the material at elevated temperatures. In the context of structural performance under extreme thermal action (such as fire), a finite element procedure is employed to analytically characterise generic temperature distributions through the thickness of a structural component arising from different types of fire exposure conditions including: a “short hot” fire leading to a high compartment temperature over a relatively short duration; and a “long cool” fire with lower compartment temperatures, but over a longer duration. Results have shown that despite the larger area under the long cool fire time-temperature curve, which traditionally represented the fire severity, the effect of the short hot fire on the nonlinear responses of beams, plates, and shallow shells is more pronounced. Also, the effect of temperature-dependent material properties is found to be more pronounced during the short hot fire rather than the long cool fire. Comparison studies have confirmed that while the current numerical and theoretical approaches for analysing of thin plates and shells are often computationally intensive, the proposed approach offers an adequate level of accuracy with a rapid convergence rate for such structures. The solutions developed can be used to: verify software used for modelling structural response to thermomechanical actions; help students and professionals appreciate the fundamental mechanics better; provide relatively quick solutions for component level analyses; and visualise internal load paths and stress trajectories in complex structural components such as composite shells that can help engineers develop deeper insights into the relevant mechanics. The formulations developed are versatile and can be used for other applications such as laminated composite or orthotropic shallow shells. A very significant by-product of developing such fundamental solutions is their potential use in the development of highly accurate hybrid elements for very efficient modelling of large problems. While this has not been fully developed and implemented in the current work, the requisite theoretical framework has been developed and reported in one of the appendices, which can be used to develop such elements and implement on an appropriate software platform.