Computational mechanics

Learning outcomes

The course aims to provide an introduction to computational issues aimed at the numerical solution of structural mechanics problems. The discretization of structural problems governed by systems of partial differential equations is considered by moving from the continuous model (infinite degrees of freedom) to the discrete model (finite degrees of freedom). Special attention is given to the Finite Element Method for the numerical solution of the main structural problems of beams and plates.

Expected learning outcomes

At the end of the course, the student is able to correctly model structural elements and structures using the finite element technique.

Course contents

1. Introduction to Computational Mechanics – Variational Approaches; Strong and Weak Formulations of the Fundamental Equations; Numerical Processing of Computer Data;
2. 1D Finite Element Method – Weak formulation of one-dimensional differential problems; Galerkin method; Interpolation of primary variables; Assembly; Boundary conditions;
3. Beam subjected to axial stress – Analytical determination of the stiffness matrix; Weak formulation; Numerical integration; Stiffness matrix for a plane truss: analytical and numerical procedure;
4. Euler-Bernoulli beam – Analytical determination of the stiffness matrix; Weak formulation; Hermite interpolation functions;
5. Timoshenko beam under bending – Weak formulation and determination of the stiffness matrix; Shear locking;
6. Numerical analysis of dynamic systems – Determination of natural frequencies; Linear dynamic analysis and Newmark methods;
7. Numerical analysis of flat frames – Euler-Bernoulli model; Timoshenko model;
8. 2D Finite Element Method – Weak formulation of 2D single-variable differential problems; Interpolation of primary variables in the plane: triangular and quadrangular elements; Mapping procedure and numerical integration; Higher-order finite elements;
9. Plane stress and strain problems – Weak formulation of the fundamental equations;
10. Bending plates – Weak formulation of the fundamental equations: Kirchhoff-Love model; Conformal and non-conformal finite elements; Weak formulation of the fundamental equations: Reissner-Mindlin model

Prerequisites

No prerequisites or propaedeutics are required. However, the course requires the knowledge of the theoretical foundations of the structural mechanics of beams and plates, in particular: beams subjected to axial stress; bending beams (Timoshenko and Euler-Bernoulli models); bent plates (Reissner-Mindlin model).

Reading/Bibliography

Reddy JN – An Introduction to the Finite Element Method. Fourth Edition, McGraw-Hill

Ferreira AJM, Fantuzzi N. – MATLAB Codes for Finite Element Analysis. Solids and Structures. Second Edition, Springer

Viola E. – Fundamentals of Matrix Analysis of Structures, Pitagora Editrice Bologna

Zienkiewicz OC – The Finite Element Method, McGraw-Hill

Teaching methods

The course program will be entirely covered during class hours.

Assessment methods

Assessment is through an oral exam, designed to assess the acquisition of the expected knowledge and skills. The oral exam consists of three questions to be answered in writing without the aid of notes or textbooks, and their discussion. The course is complemented by computer exercises that include solving structural problems using computer codes discussed in class.