Assisted design of structures

Learning objectives

The aim of the course is to provide an introduction to computational issues aimed at the numerical solution of structural mechanics problems.
We consider the discretization of structural problems governed by systems of partial differential equations through the passage from the continuous model (infinite number of degrees of freedom) to the discrete model (finite number of degrees of freedom).
In particular, the Finite Element Method is presented for the numerical resolution 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 content

1. Introduction to computational mechanics (Variational approaches; Strong and weak formulation of the fundamental equations; Numerical processing of data on the computer).

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).

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 and tools

The course program will be entirely covered during class hours.

Assessment methods and criteria

The learning assessment is done through an oral exam, aimed at ascertaining the acquisition of the expected knowledge and skills. The oral exam is based on three questions to be answered in written form without the help of notes or books, and their discussion.
The course is integrated with computer exercises that include the solution of structural problems using computer codes discussed in class.