Functionally graded materials are recognized for their ability to reduce abrupt stress transitions commonly found in laminates and are especially well-suited for use in high-temperature or severe temperature-gradient environments. Several researchers have shown significant interest in these materials and in the use of such materials to enhance structures’ responses. Nevertheless, an analysis of published works reveals that most of the research in this area has concentrated on profiles with rectangular-shaped cross-sections, regardless of whether they are plates, shells, or beams. However, beams with cross-sections commonly used in metallic construction can experience advantages if one considers a functionally graded constitution for the materials mixture evolution through the preferential bending plane. The lack of studies in this specific context has inspired the motivation for this research, which aims to explore the static and free-vibration characteristics of beams featuring unsymmetrical I-shaped cross-sections constructed from these advanced composites. To this purpose, the structures are analyzed via the finite element method. This study presents a set of parametric studies focusing on functionally graded non-symmetrical I-shaped beams, illustrating the relevance of considering such construction solution concept as a design variable option.