Analysis By The Transformed Section Method

Mechanics of Materials texts traditionally introduce composite-material members as examples to illustrate solution techniques for statically indeterminate problems. Examples of composite-material members appear in chapters that individually introduce axial loading, torsional loading, and bending. In the experience of these authors’, solutions to axial loading and torsional loading problems are presented by explicitly demonstrating application of general structural principles. That is, the principles of static equilibrium, geometric compatibility, constitutive relations, and superposition are applied in a systematic manner. Alternatively, and almost without exception, a technique commonly referred to as the transformed-section method is presented as a way to solve problems involving bending. By imposing a strain compatibility condition, materials making up the cross section are transformed into a fictitious homogeneous material. This is accomplished by adjusting the geometry of each material by a ratio of its elastic modulus to that of the base material modulus, creating a fictionalized shape of homogeneous material. The resulting single material cross section may then be analyzed in the traditional manner. In this paper the authors demonstrate that the transformed-section method can be adapted to solving composite member problems involving axial loading and torsional loading. By imposing strain compatibility conditions analogous to the bending condition a set of relationships for creating transformed sections and solving for deformation and stresses in both axial and torsion problems is developed. Further, a demonstration problem for each of these types of loading is solved by the methods developed herein. A discussion ensues as to the merits of this approach, particularly with implications that relate to student comprehension. The merit of the transformed-section method in presenting a graphical image of the relative resistance to the applied loading due to material properties is emphasized. The paper closes with a series of conclusions and recommendations for the incorporation and implementation of this alternative approach into traditional Mechanics of Materials pedagogy.