Academic Year 2016/2017 - 1° Year

ICAR/08 - 9 CFU - 1° Semester

Objectives of the course:

1. give to the students the basic knowldge of numerical methods in mechanics and of the approximations related to their use

2. give to the students the skills for performing numerical analyses of complex structures, inthe linear and non.linear range

3. give to the students the understand the principles of a numerical code of sgtructural analysis

The course includes lectures, written exercises and computer practice..

**1. METHODS OF STRUCTURAL ANALYSIS**

- Displacement method
- Variational methods. Energy principles.
- The principle of virtual works

**2. STIFFNESS MATRIX**

- Positive semidefiniteness of stiffness matrix
- Direct construction of the stiffness matrix
- Band width

**3. STRUCTURES WITH FINITE NUMBER OF DOF'S - TRUSSES**

- Assemblage of stiffness matrix
- Loads, imposed deformations and displacements: equivalent nodal forces
- Post-processing and analysis of the results
- Mass matrix

**4. VARIATIONAL METHODS OF SOLUTION FOR CONTINUOUS SYSTEMS**

- Interpolation methods. Finite differences
- Residual methods
- Ritz method
- The Ritz-Galerkin method
- The Petrov-Galerkin method

- The Finite Element Method (F.E.M.)
- Convergence and stability of the solution. Numerical issues-

**5. ANALYSIS OF CONTINUA 2D**

- The Finite Element Method for continuous systems
- Lagrangian elements
- Isoparametric elements. Numerical integration
- Equivalent nodal forces
- Post-processing. Stress evaluation and recovery
- Error estimates and Rate of convergence
- Locking issues

- Stationary problems
- Time-dependent problems. Semidiscretization

**6. FRAMES**

- Hermite shape functions
- General method for the calculation of the shape functions
- Stiffness and Mass matrices
- Equivalent nodal forces
- Post-proecssing of the resulta and errors.

**7. NON LINEAR ANALYSIS WITH F.E.M.**

- Elements of incremental analysis
- Newton's method
- Inplicit and explicit methods

- Material non linearities
- Elastic-plastic material
- Elastic-plastic trusses
- Elastic-plastic beams with concentrated hinges and with diffused plasticity
- Fundamentals of plasticity for continuous systems

- Geometric non linearities
- Geometric stiffness matrix
- Linearized stability analysis
- Incremental analysis and P-Delta effects

**8. PLATES**

- The equations of the elastic plate
- Kirchhoff-Love hupotheses
- Generalized strains and stresses
- Equilibrium equations of thin plates and boundary conditions
- Rectangular plates with various boundary conditions
- Variational solutions

- Stability of plates
- von Karman equations

- Shell finite elements
- degrees of freedom
- Interpolation of the normal
- Shear locking - mixed elements

1. J. N. Reddy – An Introduction to the Finite Element Method [Reddy] |
2. L. Corradi Dell’Acqua – Meccanica delle Strutture - Vol. 2 e Vol. 3 [MdS] |
3. Zinkiewicz – Taylor – The Finite Element Method , Vol. 1 [ZFEM] |