The development of an axisymmetric linear by high-order finite element to model filament-wound structures is presented. The primary objective of this work was to develop a ’design code' to analyze filament wound spherical pressure vessels. In order to develop a design-oriented analysis capability which can produce accurate results rather quickly with reduced input-data requirements, the total number of system equations must be reduced. To accomplish this task, a linear by high-order element was formulated which uses a single high-order displacement field finite element to model the total thickness of an axisymmetric composite structure. The displacement order for the in-plane direction remains linear, while the transverse order is user selectable. Numerical integration for stiffnesses is evaluated with respect to varying material properties and lamirna thicknesses in each individual element. Results from a computational economy study are presented showing potential time savings of 40 percent when compared to the conventional modeling scheme of using bi-linear elements. Actual test cases indicate that computation time savings may be as great as 55 percent when using linear by fourth-order elements and 45 percent when using linear by sixth-order elements. The accuracy of the element was evaluated by comparing the finite element results to elasticity solutions for isotropic, orthotropic, and filament-wound cylindrical pressure vessels. Most of the finite element results indicated a ±3 percent maximum error of the stresses compared to the elasticity results. The new linear by high order element stress results were nominally within ±2 percent of stresses calculated with conventional bilinear elements. Comparisons of finite element results for an actual filament-wound spherical pressure vessel slowed that linear by third- or fourth-order elements may be adequate for preliminary design purposes while the higher-order elements generally correlated better with the conventional bi-linear elements. Also presented is an outline of the design code and sample results for spherically wound pressure vessels. / Ph. D.