Torsional stability analysis
Introduction
Classical strength of materials is a discipline of mechanical engineering, structural engineering, civil engineering and materials engineering that studies the mechanics of deformable solids using simplified models. The resistance of an element is defined as its ability to resist applied stresses and forces without breaking, acquiring permanent deformation or deteriorating in any way.
A material resistance model establishes a relationship between the applied forces, also called loads or actions, and the stresses and displacements induced by them. Generally, geometric simplifications and restrictions imposed on the way loads are applied make the deformation and stress fields easy to calculate.
For the mechanical design of elements with complicated geometries, the resistance of materials is usually abundant and it is necessary to use techniques based on the theory of elasticity or the mechanics of more general deformable solids. These problems posed in terms of stresses and deformations can then be solved in a very approximate way with numerical methods such as finite element analysis.
Materials Strength Approach
The theory of deformable solids generally requires working with stresses and strains. These magnitudes are given by tensor fields defined on three-dimensional domains that satisfy normally complex differential equations.
However, for certain geometries that are approximately one-dimensional (beams, columns, trusses, arches, etc.) or two-dimensional (plates and sheets, membranes, etc.) the study can be simplified and can be analyzed by calculating internal forces defined on a line or a surface instead of stresses defined on a three-dimensional domain. Furthermore, the deformations can be determined with the internal forces through a certain kinematic hypothesis. In summary, for these geometries the entire study can be reduced to the study of alternative magnitudes of deformations and stresses.
The theoretical scheme of a material resistance analysis includes:
• - The kinematic hypothesis establishes what the deformations or displacement field will be like for a certain type of elements under certain types of requests. For prismatic pieces, the most common hypotheses are the Bernoulli-Navier hypothesis for bending and the Saint-Venant hypothesis for torsion.
• - The constitutive equation, which establishes a relationship between the deformations or displacements deducible from the kinematic hypothesis and the associated stresses. These equations are particular cases of the Lamé-Hooke equations.