Shape transformers for material and shape selection of lightweight beams

This paper presents a method for selecting materials, cross-section shapes, and combinations thereof. The novelty of the method is the definition of shape transformers. These parameters are dimensionless measures of the geometric quantities of a cross-section. They describe the shape properties regardless of size, and are invariant to any scaling imposed on the size of a beam cross-section in bending. Shape transformers are valuable to model the equations of mechanics and to develop selection charts for optimum design. The rationale of the approach is that the fundamental equations of continuum mechanics can be expressed by a product of four factors: the functional requirements, the material properties, the shape transformers, and the geometric quantities of a rectangle as defined by its cross-section size. This permits general expressions of performance indices to be derived for any scaling transformation. Indices for selecting materials and cross-sectional shapes that minimize the mass of beams are given for stiffness design. The last part of the paper illustrates how shape transformers facilitate a graphical exploration of performance data. The whole range of cross-sectional shapes can be visualized at a glance for each material. Lines of iso-performance enable efficiency comparison of materials and/or shapes for a given cross-section scaling. Shape transformers assist design choices and give insight into optimum selection.