# STRUCTURAL MECHANICS

*Last update 01/07/2020 18:33*

OVERVIEW

Structural Mechanics is a collection of mechanical models concerning bodies in which the shape of the body permits us to introduce simplified hypotheses that give rise to reliable and efficient solutions in applications. Firstly, we introduce the linear elasticity theory that, for many engineering applications, offers a very useful and reliable model for design. Then, invoking the set of elasticity theory equations, structural mono and bi-dimensional models are illustrated.

## AIMS AND CONTENT

LEARNING OUTCOMES

The unit is focused on the analysis of the elastic system equilibrium and strain; particularly, the course aims to study the redundant structure equilibrium, strength and stability conditions.

AIMS AND LEARNING OUTCOMES

The goal of the course is to educate the students with fundamental mechanics of solids and structures to understand with the use of different mechanical models (1D-2D- structural theories or 3-D solid mechanics) the elastic response of solid bodies made with conventional or composite materials subjected to different loading conditions. Particular attention is devoted to the phase of idealization of the model and to the discussion of results obtained with the different models.

Teaching methods

Lectures: 52 hours

SYLLABUS/CONTENT

The course is articulated in the following four parts.

1) Linear elasticity theory. The principles of stress and strain and the stress-strain relations are obtained with particular attention to the different costitutive equations for structural materials used in Yacht Design. The formulation of the Boundary Value Problem (BVP) of elasticity theory is presented for different boundary conditions (global, punctual; traction, displacement, mixed and contact). Because of the complexity of the elasticity BVP, analytical solutions to fully three-dimensional problems are very difficult to accomplish. Thus, most solutions are developed for reduced problems that include i.e. two-dimensionality to simply particular aspects of the formulation and the solution.

2) Plane problems. For the useful in many engineering applications the formulation of two-dimensional problem in elasticity is examined in details in Cartesian and coordinate systems. The two basic theories of plane strain and plane stress (and generalized plane stress) are developed and the Airy stress function solution method employed to solve a selection of explicit solutions useful in applications. The following explicit solutions are presented: rectangular domains with polynomial loading conditions, 2-D beam solution, thick-walled cylinder in pressure, stress intensity factor around the hole in a plate, Flamant problem. Exercises with Maple software.

3) Bi-dimensional theories (plates and shells). Invoking the sets of equations of elasticity the basic equations of the classical Kirchhoff’s bending theory for stiff plates are derived; the field equation in terms of displacements is used to solve plates bending problems. Navier and Levy method solutions are adopted to present explicit solutions for rectangular plates. Then, circular plate are investigated for different loading and boundary conditions. The refined Mindlin-Reissner theory in which the effects of transverse shear deformation on the bending of thick plates are take into account, is presented and, comparisons with the two models performed. The large deflection Von Karman theory is obtained to determine the critical buckling load of plates according to the equilibrium method. A brief introduction to the shell theory is illustrated. Exercises with Maple software.

4) Introduction to the Finite Element Method (FEM) for structural models. The energetic approach for the solution of the elastic problem is introduced. A brief introduction of finite element method is presented for the structural mechanics applications (phases, elements, nodes, shpe-form functions, assemblage, stiffness matrix, solution procedure). Examples with Maple software.

RECOMMENDED READING/BIBLIOGRAPHY

Corradi Dell’Acqua, L., Meccanica delle strutture 2, McGraw-Hill, London (2010).

Nunziante, L., Gambarotta, L., Tralli, A., Scienza delle Costruzioni, McGraw-Hill (2008).

Mase, G.T. Mase, G.E., Continuum Mechanics for Engineering, CRC Press, New York (1999).

Sadd, M.H., Elasticity: Theory, Applications, and Numerics, Elsevier (2014).

## TEACHERS AND EXAM BOARD

Exam Board

ROBERTA SBURLATI (President)

ROBERTO CIANCI

## LESSONS

Teaching methods

Lectures: 52 hours

ORARI

L'orario di tutti gli insegnamenti è consultabile su EasyAcademy.

## EXAMS

Exam description

The exam is oral with two/three questions concerning the different parts of the program.

A laboratory in which an elastic plane solution problem and/or a plate elastic solution are obtained using the Maple software is proposed to the students during the course (optional, presence is required).

Web-support: notes and slides in aulaweb (slides in English and notes in Italian).

Assessment methods

The exam is oral with two/three questions concerning the different parts of the program.