The course aim at:

- describing in detail the linear elastic problem as the fundamental problem governing the behaviour of solids and structures;

- formulating the linear elastic problem for solids and structures in a way suitable for numerical treatment;

- describing the Finite Element Method for linear elastic bodies and structures;

- giving to the students capabilities in the implementation of the Finite Element Method through laboratory sessions;

- describing the elasto-plastic behaviour of materials and structures;

- presenting the theory of limit analysis for elasto-plastic continua and structures;

- presenting the theory of limit states occurring due to variable-repeated loads acting on elasto-plastic continua and structures.

After attending the course and after the final examination, the student will:

- know the formulation and relevant hypotheses of the linear elastic problem for solid continua and structural elements;

- know the general formulation of the Finite Element Method for linear elastic problems;

- know how to implement in computer codes the Finite Element Method for linear elastic continua and structures;

- know the formulation of elasto-plastic constituive laws;

- know the theory of limit analysis for elasto-perfectly plastic continua and structures;

- be able to apply the theory of limit analysis for the determination of the collapse load of bi-dimensional elasto-plastic frame structures;

- know the theory of limit states for elasto-plastic bodies and structures subject to variable, repeated loads.

The student will be able to discuss the above acquired knowledges with a clear and appropriate language and will be able to reproduce the fundamental reasoning and proofs that bring to the demonstration of the theorems presented during the classes.

- Part 1: The structural problem.

General formulation of the elastic boundary-value problem.

Virtual work principle; virtual displacements and virtual forces principles.

Potential energy and complementary energy theorems.

Plane strain and plane stress elastic problems.

Axi-symmetric elastic problem.

Structural theories: Timoshenko and Euler-Bernoulli beam theories.

- Part 2: The finite element method in linear problems

Displacement approach for truss structures.

The Rayleigh-Ritz method for the displacement approach.

General formulation of the displacement-based finite element method: mesh construction; displacement model; stiffness and mass matrices; equivalent nodal loads; assembling procedure; solution; convergence criteria.

Finite elements for plane problems.

Isoparametric finite elements.

Finite elements for axi-symmetric problems.

Finite elements for Euler-Bernoulli and Timoshenko beams.

Numerical integration.

- Part 3: Elasto-plastic behaviour of materials and beams

Uni-dimensional elasto-plastic constitutive law: perfectly-plastic and hardening behaviours.

Three-dimensional elasto-plastic constitutive law: von Mises yielding function, associative flow rule, isotropic hardening.

Perfectly-plastic beams under bending and axial force.

Limit domains for beam sections under bending and axial force.

- Part 4: Limit analysis

Fundamentals of limit analysis for frames: static and kinematic theorems, corollaries.

Limit analysis of frames by direct methods and by linear programming.

Limit analysis for elasto-plastic continua.

Shakedown analysis: Bleich-Melan theorem.

Requested background

Mathematical analysis

Rigid body Mechanics

Deformable Solids and structural mechanics

The examination includes the following parts.

- An (elective) work in which the student applies the knowledge acquired during the laboratory classes on the Finite Element Method.

- A written exam including quantitative exercises and open questions on the course topics.

- An oral exam in which the examiner check the student's preparation asking supplementary information on the solved quantitative exercises of the written text and ask general questions on the course contents.

The student during the final oral exam must be able to clearly describe and critically discuss the topics, with the related hyoptheses, the critical points, the mechanical meaning and their consequences.