Thin plates, sheets and shells are an important category of structural components, and while commonly used, they are often only utilised in simplistic geometries. An increased understanding of the geometric and structural behaviour of more sophisticated geometries allows for the optimisation of performance. This thesis considers a thin plate which has (orhas been given) out-of-plane depth or texture: a dimpled sheet. By developing new methods of analysis for these types of modified plate geometries, increases in performance can be sought and quantified. Compared to flat plates, corrugated sheets have an increased bending stiffness in one direction, whilst dimpled sheets have an increase in bending stiffness in both orthogonal directions. Comparing dimpled to corrugated sheets, a “second moment of area” approach might be used; however, this thesis will show that such an analysis is unsuitable. Instead a wholly new approach is developed to describe and quantify the structural behaviour of a dimpled sheet, with the key observation being the treatment of each dimple as an elastic inclusion. Theoretical analysis is carried out which confirms the applicability of representing a single dimple as an elastic inclusion, and which quantifies the relationship between dimple geometry and the effective stiffness of the inclusion. The applicability of this representation is also confirmed though use of relevant finite element analysis. Analysis of the overall performance of a plate with a pattern of inclusions is subsequently carried out. A theoretical formula is derived that accurately predicts the smeared overall elastic modulus of an inclusion patterned plate, and the suitability of this formula is backed-up by extensive finite element analysis. This formula also compares favourably to existing “rule-of-mixtures” approaches, although it is superior to existing rules due to its incorporation of Poisson’s ratio terms. Practical experiments on perforated strips explore the behaviour of plates which have inclusions of zero stiffness, with favourable agreement to the derived theory. By combining the analyses of the previous sections, the overall performance of a dimpled sheet is investigated. Making suitable adjustments to the effective inclusion representing each dimple, due to the proximity of adjacent dimples, a complete theoretical prediction of the structural performance of a dimpled sheet is derived. Finite element analysis is used to validate that the theoretical model is suitable for predicting and accurately capturing both the increase in bending stiffness of a dimpled sheet, as well as the reduction in stretching stiffness. Furthermore, practical experiments on physical dimpled sheet specimens confirm the increased bending stiffness which is obtained from dimpled sheets, as compared to identical specimens without dimpling.