Strength of materials

Resistance of materials (RoM) is a special discipline of continuum mechanics, used to calculate stresses and strains 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 linked to a frame and subjected to external mechanical loads.

RoM allows us to reduce the study of a structure's overall behavior (the relationship between stresses - forces or moments - and displacements) to that of the local behavior of the materials making it up(the relationship between stresses and strains). Mechanical stresses can be seen as " cohesive forces " in matter. The deformations of a physical object can be observed by a variation in its dimensions or overall shape.

The use of my RoM has two main objectives:

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

2) Dimensioning mechanical structures. To do this, we need to knowthestate of stresses and/orstrains within these structures. This is made possible by RdM modeling, which enables us to predict these stress/strain states.

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

Prerequisites* :

Rigid solid mechanics

Trigonometry

Notions of derivatives and primitives of functions

Recommended prerequisites* :

Analysis and linear algebra in L1+L2

Final exam with CC and max rule + practical work

This begins with a brief reminder of the Fundamental Principle of Statics (PFS) to determine the connection forces of a beam to its frame, followed by the calculation of forces within a lattice of bars, up to and including the determination of the displacement of the lattice nodes.

Next, the "cut-off" method for determining internal forces is introduced. After introducing the internal forces in a beam, the local equilibrium equations are established, along with the method for solving them. Simple load cases are studied: tension/compression, bending and torsion. Following the introduction of the notion of deformation, the behavior laws of beams are defined and their use to determine 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 present energy methods, which are an application of the Virtual Power Principle.