AIMS AND LEARNING OUTCOMES

The course introduces students to the Solid Mechanics through the formulation of appropriate mathematical models able to simulate the real behavior of materials. Among these the simplest and most widely used is known as Cauchy continuum. Therefore, this course represents an introduction to the Mechanics of Materials (to use a terminology of English origin) which is fundamental in every field of Engineering, not only for Civil and Environmental Engineers. The main learning outcome of the course is to provide, in its first part, fundamentals on the analysis of tension, deformation and constitutive law of three-dimensional solids in the linear field (i.e., small displacements and displacement gradients). The second part of the course deals with a particular continuous model able to represent a continuous mono-dimensional beam type (i.e., the De Saint Venant model), whose solution allows to deal with all the stress cases present in the structural engineering problems. The strong connection with Module 1 is repeatedly underlined with simple examples. At the end of the course the students will be able to design and verify simple beams subject to stress not only of bi-dimensional but also of three-dimensional nature (as in the case of biaxial flexure and torsion).

During the course the student must develop the following main skills as a result of the learning process:

- physically interpreting deformation and stress states
- determining principal deformations and stresses and their corresponding directions in both analytical and graphical approach
- writing compatibility and equilibrium (balance) equations of the continuous body
- using the constitutive law equations of isotropic linear elastic materials and write the potential elastic energy
- to determine stress, strain and displacement fields deriving from different simple and compound stresses in the De Saint Venant continuous model: axial load, bending and biaxial flexure, combined bending and axial loading, torsion, transverse shear, combined torsion and shear stresses
- graphically showing the stress fields determined
- using approximate theories for the evaluation of uniform torsion in open and closed thin-walled sections
- performing verifications of strength in the presence of multiaxial stress states
- being able to deal with design and verification problems for simple beams

TEACHING METHODS

Lectures: on the blackboard using slide projections

SYLLABUS/CONTENT

The Cauchy continuum

  Deformation analysis
  Stress analysis
  Virtual Work Identity
  Constitutive law for isotropic material
  The elastic problem and the De Saint Venant principle

The De Saint Venant problem

    Semi-inverse method
    General solution for normal stress
    Sub-problem 1: uniform extension
    Sub-problem 2: uniform bending
    Mixed problems (biaxial bending and eccentric normal force)
    Sub-problem 3: uniform torsion
    Sub-problem 4: non-uniform bending
    Mixed problems: transverse shear and torsion, shear center
    Notes on resistance criteria and equivalent (ideal) stress

RECOMMENDED READING/BIBLIOGRAPHY

Course notes available on Aulaweb

Nunziante, Gambarotta, Tralli - Scienza delle Costruzioni - McGraw Hll (capitoli 5 e 6)
Luongo, Paolone - Scienza delle Costruzioni: Il continuo di Cauchy - Casa Editrice Ambrosiana (capitoli 1-6, 10)
Casini, Vasta - Scienza delle Costruzioni - Città Studi Edizioni (capitoli 13-22, 24)