Periodically perforated sheets(PS) are ubiquitous in nature as well as in engineered artifacts developed for aerospace, automotive, marine, nuclear and structural applications. PS are indispensable for saving weight and cost for aircraft; for enhancing safety and integrity of heat exchangers used in nuclear and thermal power stations. Ancient PS grills and lattice frames dating back to 1000 BC continue to inspire contemporary art and architecture, buildings and furniture. PS design and analysis, however, is a complex affair stemming from the inherent configurational anisotropy induced by periodicity. In addition, complex boundary conditions complicate the analysis. Unlike atoms in crystalline media, both shape and periodicity of perforations control this anisotropic nature. This thesis explores theoretical and numerical strategies for evaluating the effective anisotropic elastic moduli of PS. Following an experimental prelude for visualizing the PS stress field in a photoelastic sheet and a brief review of PS theory, this thesis proposes a novel theoretical numerical hybrid method for determining the Airy stress function constants. The proposed hybrid method can be exploited experimentally using automated vision based imaging technologies to measure the boundary displacements noninvasively. For determining the Airy constants periodic boundary conditions to the unit cell are applied, the displacement components around the PS hole boundary are obtained using FEM. Using these constants the PS stress field is reconstructed to assess the efficacy of the proposed hybrid method. It is observed that in general while the actual and the reconstructed stress fields agree reasonably well, more refined boundary data obtained either numerically or experimentally can enhance the accuracy further. The thesis then makes an extensive presentation of anisotropic moduli in a variety of PS designs configured on rectangular or square layouts. Conventional as well as some exotic patterns with cusps and satellite holes are examined, and the results are presented graphically to aid the designer. Finally, some special topics pertaining PS design and analysis are discussed to help overcome the inherent limitations of solutions based on applying periodic boundary conditions. In this vein, strategies for achieving a functionally graded PS are presented by altering the pitch and hole size. These strategies assume importance near boundaries as well as near concentrated forces inducing stress gradients. Other special topics include the applicability of tensor transformation rule to PS anisotropy. The effective bulk modulus which remains a scalar invariant is exploited to assess the validity of tensor transformation in a square PS. The rule of mixture widely used in homogenization of composite media is also discussed briefly. Thus, this thesis makes an attempt to demonstrate the power of blending micromechanics with experiments and FEM to aid in PS design and analysis.