Wrinkling of thin membranes due to different in-plane loading and boundary conditions has drawn a lot of attention among researchers in the field of structural engineering since the development of tension field theory for the designs of thin webs for early aircraft structures. More recently, prestressed lightweight membrane structures have been proposed for future space missions, for example solar sails, the Next Generation Space Telescope sunshield and some space-based radar systems. These structures are often partially wrinkled during operation. The formation of wrinkles alters the load paths and structural stiffness of the membranes. More importantly its occurrence degrades the surface accuracy of these structures, which is a key design parameter.

This dissertation provides two methods to predict the details of wrinkles, namely wrinkle wavelength and amplitude in these membrane structures. Both of these proposed methods exploit the key observation, shown by physical models, that the main parameter which governs the wrinkle details is the bending stiffness of the membrane.

The first method models the membrane using thin shell elements
available in the commercial non-linear finite element code
**ABAQUS**. The analysis uses a buckling prediction analysis
to obtain the initial imperfections that, once introduced in the
structure, would induce the formation of wrinkles. This analysis
predicts the final wrinkle shapes and the out-of-plane deformation
of the membrane, which quantify the wrinkle details. Compressive
stresses are allowed to develop in this model.

A simple analytical wrinkle model is the proposed second method, based on the assumption that a membrane is able to resist a small compressive stress once it has wrinkled. This critical wrinkling stress is a function of the bending stiffness and wrinkle wavelength. The wrinkle amplitude can be predicted by considering the total strain in the membrane, as the sum of two components, a material strain and a wrinkling strain.

Two membrane structures subjected to in-plane loads are investigated. The first structure is a classical rectangular membrane under shear. The second structure is a square membrane loaded with tension forces at its four corners. In the first case, the wrinkle pattern consists of primarily of wrinkles at 45 degrees and the state of stress in the membrane can be readily determined by tension field theory. The predicted wrinkle wavelength is inversely proportional and the amplitude is directly proportional to the imposed shear angle. In the second case, two wrinkling regimes are identified by varying the corner load ratio. The first regime occurs for symmetric and moderately asymmetric loading; it is characterised by small, radial corner wrinkles. The second regime occurs for strongly asymmetric loading with the formation of a single, large diagonal wrinkle in addition to small radial corner wrinkles. Several simple equilibrium, no-compression stress fields are proposed to capture the behaviour of this wrinkled membrane. Analytical solutions for predicting the wrinkle details are hence derived.

The analytical predictions are validated against experimental measurements and detailed finite element simulation results. It is shown that thin shell finite element simulations can provide very detailed and accurate predictions of wrinkle details, although they are computationally expensive. The simple theory provides an alternative solution to obtain this information and is useful for preliminary design of thin membrane structures.

*Keywords*: wrinkling; membranes;
tension field; finite elements; thin shell; critical wrinkling
stress; wrinkle wavelength; wrinkle amplitude.