Analytic shapes of air-filled thin-walled membranes impacted upon a variable transmural pressure

This work presents analytical solutions for the shapes of air-filled, thin-walled membranes subjected to variable transmural pressures. These elastic membranes, resting on a rigid foundation, are assumed to be in equilibrium. The governing equations are derived from a static balance of horizontal and vertical operating forces acting on the membrane tube's surface. Unlike previous studies that assumed constant transmural pressure (acting normal to the wall surface), we generalize it to account for variable net force differences due to external and internal pressure forces. This allows us to analytically determine the corresponding shapes of the membranes and derive closed-form expressions for their mechanical properties. Notably, we demonstrate that the air pressure inside the tube can be reduced below atmospheric pressure while maintaining inflation, where the external pressure actually corresponds to a pulling from the outside.