6 credits during the 2nd year of 8694 Sciences of architecture (L-17) GENOVA

**Sciences of architecture 8694 (coorte 2020/2021)**- MATHEMATICS 1 56394

## AIMS AND CONTENT

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)

## TEACHERS AND EXAM BOARD

**Office hours:** Tuesday afternoon from 15 onwards (I semester), from 16 onwards (II semester)
Office hours on request (appointment via email)

Exam Board

LUIGI GAMBAROTTA (President)

ANDREA BACIGALUPO

GIUSEPPE PICCARDO (President Substitute)

## LESSONS

TEACHING METHODS

Lectures: on the blackboard using slide projections

LESSONS START

Class schedule

All class schedules are posted on the EasyAcademy portal.

## EXAMS

EXAM DESCRIPTION

The exam consists of a written and an oral test.

Furthermore, during the Course, there will be two written tests on the first (Cauchy continuum) and the second part (DSV problem) of the Course. The students that will pass them, will be enabled to access directly the oral exam.

ASSESSMENT METHODS

The written test is based on the solution of some (2/3) problems similar to those dealt with during the course exercises, concerning the whole program of the course. The use of books or notes, and electronic equipment of any kind (cell phones, tablets, laptops) is not permitted. The written tests are valid during the exam session in which they are sustained (June-September or January-February). Students who have obtained a grade equal to or greater than 18 in the written test are admitted to the oral exam. Handing in a test cancels the outcome of any previous written tests.

The written partial tests proposed during the course (II semester), with an average grade equal to or greater than 18, replace the written test. The partial tests lose validity if the oral test is not sustained by the end of the Session in which they are held (September). The topics of the partial tests concern the contents developed in the periods preceding the test itself and concern the first (Cauchy continuum) and the second part (DSV model solutions) of the course. Handing in a test cancels the outcome of the partial tests.

The oral examination is aimed at verifying the acquisition of the concepts and analytical skills acquired by the student and the knowledge of the theoretical part of the program, as well as a possible discussion of the written test. To take the oral test it is necessary to have a valid written test and to be in compliance with the prerequisites. If the outcome of the oral test is not sufficient, a new written exam must be taken.

The exams take place in Italian. The oral exam can be taken in English.

Exam schedule

Date | Time | Location | Type | Notes |
---|---|---|---|---|

21/12/2021 | 15:00 | GENOVA | Scritto | |

07/01/2022 | 15:00 | GENOVA | Orale | |

20/01/2022 | 15:00 | GENOVA | Scritto | |

11/02/2022 | 15:00 | GENOVA | Orale | |

21/06/2022 | 15:00 | GENOVA | Scritto | |

04/07/2022 | 15:00 | GENOVA | Orale | |

18/07/2022 | 15:00 | GENOVA | Scritto | |

01/08/2022 | 15:00 | GENOVA | Orale | |

02/09/2022 | 15:00 | GENOVA | Scritto | |

12/09/2022 | 15:00 | GENOVA | Orale |