Academic Year 2019/2020 - 1° Year

ICAR/08 - 9 CFU - 2° 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, in the linear and non.linear range.

3. give to the students the ability to understand the basic principles of a numerical code of structural analysis, in order to use it with awereness.

The course includes lectures, written exercises and computer practice.

The course is copleted in 2 semesters, and is divided in 2 modules. The first module includes about 28 hours of classes (3 credits), and covers points 1-4 pf the programme. At the end there will be an intermediate examination that will determine part of th efinal grade. The second module, in the spring semester, covers the remaining points o fthe programme, and includes numericla practices and the use of computer structura codes. .

The course in taight in two semestres. In the first semester (27 hours) the students receive the basic notions of structural methods of analysis, with reference to thd displacement method for finite degrees of freedom structures, and basic notions of one-dimensional plasticity.

In the second semester it is treated the Finite Element Method for ocntinuous structures (2d and 3D) and non linear methods of analysis.

**1. METHODS OF STRUCTURAL ANALYSIS**

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

**2. STIFFNESS AND MASS MATRICES**

- Direct construction of the stiffness matrix and of the mass matrix. Mechanical interpretation.
- Positive semidefiniteness 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. INTRODUCTION TO MATERIAL NON LINEARITIES**

- Plasticity theory for 1D systems.
- Yield domain, hardening, residual strain, dissipation.
- Incremental and cycling loading of structures composed by 1D elasto-plastic elements.
- Plastic bendong of beams.
- Plastic hinges. Incremental and cycling loading of beams subjected to bendong only.

**5. 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-

**6. 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

**7. FRAMES**

- Hermite shape functions. Continuity requirements.
- General method for the calculation of the shape functions.
- Higher order beam models.
- Stiffness and Mass matrices
- Equivalent nodal forces
- Post-proecssing of the resulta and errors.

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

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

- Material non linearities
- Fundamentals of plasticity for continuous systems. Drucker's postulate. Associated and non associated plasticity.
- Computatioal analysis of plastic deformation. The return mapping algoriithm.
- A framework for material non linear analysis of a continuous structure.
- Elastic-plastic beams with concentrated hinges and with diffused plasticity.

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

**9. 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 Mc Graw Hill [Reddy]

2. L. Corradi Dell’Acqua – Meccanica delle Strutture - Vol. 2 e Vol. 3 [MdS]

3. Zinkiewicz – Taylor – The Finite Element Method , Vol. 1 Butterworth [ZFEM]

4. Eugenio Oñate - STRUCTURAL ANALYSIS WITH THE FINITE ELEMENT METHOD. Volume 1 : The Basis and Solids. Volume 2 . Beams, Plates and Shells. Springer.[ONA}

5. J. Lublineer - Plasticity Theory. Mc Millan [LUB]