Towards a Conic Programming Implementation of Tension Field Theory for the Static Analysis of Pneumatic Membrane Structures

Abstract

This dissertation addresses the suitability, applicability, and stability of finite element membrane formulations for modeling and optimizing pressurized membranes subjected to external fluid loading. Pressurized membranes are pressurized thin structures that lack out-of-plane stiffness, withstand external loading through tensile stresses only, and whose modeling and design is complicated by multiple nonlinearities: finite deformations, material nonlinearities, deformation-dependent loads, and wrinkling. The suitability of membrane formulations over shell formulations for modeling pressurized thin structures has lacked a clear theoretical basis, while existing pressurized membrane formulations have neglected to exploit convexity properties to mitigate the numerical instabilities caused by the lack of out-of-plane stiffness and wrinkling.

A pressurized membrane’s capacity to withstand transverse loads is affected by three key design variables: geometry, thickness, and internal pressure. To advance pressurized membrane modeling and design, in this dissertation, state-of-the art computational methods are used in parametric and optimization studies to investigate the effects of these design variables. These computational methods are then developed further, resulting in a novel approach for modeling the wrinkling of membrane structures.

A parametric exploration of the design space is presented first, considering both shell and membrane finite element formulations. The effects of these design variables on the mechanical response are demonstrated by the obtained original results, and conditions that result in membrane-dominated responses are identified. To investigate the effects of these design variables further, an optimization methodology based on the adjoint method is developed and applied to pressurized membrane design. In these parametric and optimization studies, it is observed that wrinkling constitutes a significant hindrance to the suitability of the membrane formulation, and that satisfactory options for handling this complication are lacking. In response, this dissertation proposes a wrinkling model for pressurized membranes based on conic programming. The method, which offers unparalleled robustness and efficiency due to the benefits of convexity, is demonstrated with ample numerical examples. Altogether, the dissertation demonstrates that pressurized thin structures subject to external fluid loading can be effectively modeled and optimized using membrane formulations, and that such formulations are worthy of continued research.