Strength of Materials

Materials strength (RdM) is a specific branch of continuum mechanics that enables the calculation of stresses and strains in slender structures made of various materials (machinery, mechanical engineering, building, and civil engineering). It involves a 1D static model of a deformable solid treated as a beam connected to a frame and subjected to external mechanical loads.

The RdM makes it possible to reduce the study of a structure’s global behavior (the relationship between loads—forces or moments—and displacements) to that of the local behavior of its constituent materials (the relationship between stresses and strains). Mechanical stresses can be viewed as the “cohesive forces ” within the material. The strains of a physical object are observed as changes in its dimensions or overall shape.

The use of my RdM serves two main purposes:

1)Provide a generaldescription of themechanical behavior of a solid under small deformations during use.

2) Design mechanical structures. To do this, it is necessary to knowthe stress and/orstrainconditions within these structures. This is made possible by FEM modeling, which allows us to predict these stress and strain conditions.

This course is therefore an application of Hooke’s linear isothermal elasticity in the 1D case. The mechanical structures considered here are thus represented by beams—that is, one-dimensional elastic media that are, a priori, curved.

Required prerequisites*:

Rigid Body Mechanics

Trigonometry

Concepts of Derivatives and Antiderivatives of Functions

Recommended prerequisites*:

Analysis and Linear Algebra of L1+L2

Final Exam with Multiple Choice Questions and the Maximum Rule + Practical Assignment

This begins with a brief review of the fundamental principle of statics (FPS) to determine the forces at the connections between a beam and its support structure, followed by the calculation of forces within a truss, and concludes with the determination of the displacements of the truss nodes.

Next, the so-called “section” method is introduced to determine internal forces. After introducing the internal forces in a beam, the local equilibrium equations are established, along with the method for solving them. Various cases of simple loading are examined: tension/compression, bending, and torsion. Following the introduction of the concept of deformation, the laws governing beam behavior are defined, and their application is developed to determine the deformation of a beam.

Before highlighting the connection between this approach and the 3-D elasticity of deformable media (next semester), we will discuss energy methods, which are an application of the principle of virtual powers.