Strength of materials

Material resistance (RdM) is a specific discipline of continuum mechanics that enables the calculation of stresses and deformations in slender structures made of different materials (machinery, mechanical engineering, building, and civil engineering). It involves 1D static modeling of a deformable solid assimilated to a beam connected to a frame and subjected to external mechanical stresses.

RdM allows the study of the overall behavior of a structure (relationship between stresses—forces or moments—and displacements) to be reduced to that of the local behavior of the materials composing it (relationship between stresses and strains). Mechanical stresses can be seen as the " cohesive forces " of the material. The deformations of a physical object can be observed as a change in its dimensions or overall shape.

The use of my RdM has two main objectives:

1)Characterize the overallmechanical behavior of a solid under small deformations.

2) Dimensioning mechanical structures. To do this, it is necessary to knowthe state of stress and/ordeformation within these structures. This is made possible by RdM modeling, which can predict these states of stress/deformation.

This teaching is therefore an application of linear isothermal Hooke's law in the 1D case. The mechanical structures considered are thus represented by beams, i.e., one-dimensional elastic media, which are a priori curved.

Required prerequisites*:

Rigid body mechanics

Trigonometry

Concepts of derivatives and primitives of functions

Recommended prerequisites:

Analysis and Linear Algebra L1+L2

Final exam with CC and max rule + TP

It begins with a brief review of the fundamental principle of statics (FPF) to determine the forces acting on the connections between a beam and its frame, followed by the calculation of forces within a truss, leading to the determination of the displacement of the truss nodes.

Next, the "cut method" is introduced to determine internal forces. After introducing internal forces in a beam, local equilibrium equations are established, along with the method for solving them. Various simple load cases are studied: tension/compression, bending, and torsion. Following the introduction of the concept of deformation, the laws of beam behavior are defined and their use in determining the deformation of a beam is developed.

Before highlighting the link between this approach and the 3-D elasticity of deformable media (next semester), we will discuss energy methods, which are an application of the Virtual Power Principle.