Assisted design of structures

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).

Aims

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.

Description

Contents

1. Introduction to computational mechanics
   Variational approaches
   Strong (differential) and weak (integral) formulation of the fundamental equations
   Numerical processing of computer data
2. Finite Element Method: one-dimensional case
   Weak formulation of one-dimensional differential problems
   Galerkin method
   Interpolation of primary variables
   Assembly procedure
   Imposition of boundary conditions
3. Beam subjected to axial stress
   Analytical determination of the stiffness matrix
   Weak formulation of the problem
   Numerical integration
   Stiffness matrix for a plane truss: analytical and numerical procedure
4. Bent beam: Euler-Bernoulli model
   Analytical determination of the stiffness matrix
   Weak formulation of the problem
   Hermite interpolating functions
5. Bent beam: Timoshenko model
   Weak formulation of the problem and determination of the stiffness matrix
   Shear Locking
6. Numerical analysis of dynamical systems
   Finite Element Model for the determination of natural frequencies
   Linear dynamic analysis and Newmark methods
7. Numerical analysis of flat frames
   Finite element model for flat frames: Euler-Bernoulli model
   Finite element model for flat frames: Timoshenko model
8. Finite Element Method: two-dimensional case
   Weak formulation of two-dimensional single-variable differential problems
   Interpolation of primary variables in the plane: triangular and quadrangular elements
   Mapping procedure and two-dimensional numerical integration
   Finite elements of higher order
   Criticality of Finite Element modeling
9. Plane stress and strain problems
   Weak formulation of the fundamental equations for plane elasticity problems
10. Inflected plates
    Weak formulation of the fundamental equations: Kirchhoff-Love model
    Conforming and non-conforming finite elements
    Weak formulation of the fundamental equations: Reissner-Mindlin model

Teaching methods:

The course program is entirely carried out during class hours.

Examination methods

The Computational Mechanics exam includes an oral test and the verification of the exercises carried out during the course.

REFERENCES

• Reddy JN – An Introduction to the Finite Element Method. third edition, McGraw-Hill
  
• Ferreira AJM – MATLAB Codes for Finite Element Analysis. solids and structures, Springer
  
• Viola E. – Fundamentals of Matrix Analysis of Structures, Pythagoras Publishing Bologna
  
• Zienkiewicz OC – The Finite Element Method, McGraw-Hill