Tanks are storage vessels for liquids. They can have different appearances; some are short and wide, others are tall and slim, some are small, others are large. In this thesis a tank of 6 m in both diameter and height has been used to obtain numerical results of the stresses in the tank. Tanks are most often thin-walled with stepwise variable shell thickness with thicker wall sections at the bottom of the tank and thinner at the top. Since they are thin-walled they are susceptible to buckling and there are conditions the shell construction must meet. The conditions that has to be met are determined by the laws and regulations that govern tank design. The National Board of Housing, Building and Planning (Boverket) is the new Swedish authority for rules of tank design and the Eurocodes are the new family of standards that should be followed. Sweco Industry AB is the outsourcer of this thesis and wants to clarify what rules that apply now when the Eurocodes are to be followed. The thesis project has produced a calculation document in Mathcad for tank shell design according to the Eurocodes with stress calculations according to membrane theory and linear elastic shell analysis. This thesis has also produced a comparison of stresses calculated using membrane theory, linear elastic shell analysis and finite element method (FEM). The comparison has been made for numerical results given for an arbitrarily designed tank wall. The loads acting on the tank included in the description were self-weight, internal and hydrostatic pressure as well as wind and snow loads. The loads were described in accordance with the Eurocodes. Some assumptions had to be made where the standard was vague or deficient in order to make calculations by hand possible. For example, the wind load had to be described as an axisymmetrically distributed load rather than an angularly varying. The stresses in the tank wall were calculated through creating free-body diagrams and declaring equations for force and moment equilibrium. The loads and boundary conditions were set in a corresponding manner in the FEM software Ansys as in the calculation document in order to obtain comparable results. When compared, the stress results calculated with membrane theory and FEM were quite similar while the stresses calculated with linear analysis were a lot larger. The bending moments were assumed to be too large which make the results of the linear analysis dominated by the moments. The arbitrarily dimensions set for the tank did thus not fullfill the conditions when linear analysis was used but did so for membrane theory and FE-analysis. Since the results calculated with membrane theory were very close to FEM in most cases, even without expressions for local buckling, it was assumed to be an adequate method in this application. Expressions for local buckling are although needed for the meridional normal stress. The conclusions of the results obtained are that membrane theory is a simple and adequate method in most cases. Linear analysis thus becomes redundant since it is more complicated and more easily leads to faulty results. Furthermore it cannot be used for higher consequence classes than membrane theory. FEM, with a computer software such as Ansys, is although the most usable calculation method since it can conduct more complicated calculations and is allowed to be used for all consequence classes.

This work aims at dimensional reduction of nonlinear material models in an asymptotically accurate manner. The three-dimensional(3-D) nonlinear material models considered include isotropic, orthotropic and dielectric compressible hyperelastic material models. Hyperelastic materials have potential applications in space-based inflatable structures, pneumatic membranes, replacements for soft biological tissues, prosthetic devices, compliant robots, high-altitude airships and artificial blood pumps, to name a few. Such structures have special engineering properties like high strength-to-mass ratio, low deflated volume and low inflated density. The majority of these applications imply a thin shell form-factor, rendering the problem geometrically nonlinear as well. Despite their superior engineering properties and potential uses, there are no proper analysis tools available to analyze these structures accurately yet efficiently. The development of a unified analytical model for both material and geometric nonlinearities encounters mathematical difficulties in the theory but its results have considerable scope. Therefore, a novel tool is needed to dimensionally reduce these nonlinear material models. In this thesis, Prof. Berdichevsky’s Variational Asymptotic Method(VAM) has been applied rigorously to alleviate the difficulties faced in modeling thin shell structures(made of such nonlinear materials for the first time in the history of VAM) which inherently exhibit geometric small parameters(such as the ratio of thickness to shortest wavelength of the deformation along the shell reference surface) and physical small parameters(such as moderate strains in certain applications). Saint Venant-Kirchhoff and neo-Hookean 3-D strain energy functions are considered for isotropic hyperelastic material modeling. Further, these two material models are augmented with electromechanical coupling term through Maxwell stress tensor for dielectric hyperelastic material modeling. A polyconvex 3-D strain energy function is used for the orthotropic hyperelastic model. Upon the application of VAM, in each of the above cases, the original 3-D nonlinear electroelastic problem splits into a nonlinear one-dimensional (1-D) through-the-thickness analysis and a nonlinear two-dimensional(2-D) shell analysis. This greatly reduces the computational cost compared to a full 3-D analysis. Through-the-thickness analysis provides a 2-D nonlinear constitutive law for the shell equations and a set of recovery relations that expresses the 3-D field variables (displacements, strains and stresses) through thethicknessintermsof2-D shell variables calculated in the shell analysis (2-D). Analytical expressions (asymptotically accurate) are derived for stiffness, strains, stresses and 3-D warping field for all three material types. Consistent with the three types of 2-D nonlinear constitutive laws,2-D shell theories and corresponding finite element programs have been developed. Validation of present theory is carried out with a few standard test cases for isotropic hyperelastic material model. For two additional test cases, 3-Dfinite element analysis results for isotropic hyperelastic material model are provided as further proofs of the simultaneous accuracy and computational efficiency of the current asymptotically-correct dimensionally-reduced approach. Application of the dimensionally-reduced dielectric hyperelastic material model is demonstrated through the actuation of a clamped membrane subjected to an electric field. Finally, the through-the-thickness and shell analysis procedures are outlined for the orthotropic nonlinear material model.