COMPUTATIONAL MECHANICS

Objectives of the course:

1-Knowledge and understanding: give to the student the basic knowledge and understanding underlying numerical methods in structural mechanics, enabling an informer grasp of the approximations.

2-Applying knowledge and understanding: give to the student the skills for performing numerical analyses of complex structures, in the linear and non linear range.

3-Making judgements: to equip the student with the ability to gather and interpret results obtained from structural codes, ensuring their correct use.

4-Comunication skills: to equip the student with solid communication skills regarding the topics covered, enabling them to work independently within work groups.

5- Learning skills: to provide the student with the learning skills necessary to independently undertake and develop in-depth studies of the topics covered.

The course includes lectures, written exercises and computer practice.

The course is completed in 1 semesters,  and is divided in 2 modules. The first module includes about 28 hours of classes (3 credits), and covers points 1-4 of the program. At the end there will be an intermediate examination that will determine part of the final grade. The second module, covers the remaining points of the program, and includes numerical practices and the use of computer codes for structural analysis.

Classes will be held in person; the theory will be followed by practical examples.

The student must have a good knowledge of basic structural mechanics. Furthermore, a knowledge of structural dynamics and of concrete structures is useful but not mandatory.

Attendance is mandatory for admission to the oral exam.

1. METHODS OF STRUCTURAL ANALYSIS

1. Displacement method.
2. Variational methods, Energy principles.
3. The principle of virtual works.

2. STIFFNESS AND MASS MATRICES

1. Direct construction of the stiffness matrix and of the mass matrix. Mechanical interpretation.
2. Positive semidefiniteness of the stiffness matrix.
3. Band width.

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

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

4. INTRODUCTION TO MATERIAL NON LINEARITIES

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

5. VARIATIONAL METHODS OF SOLUTION FOR CONTINUOUS SYSTEMS

1. Interpolation methods. Finite differences
2. Residual methods
3. Ritz method
   1. The Ritz-Galerkin method
   2. The Petrov-Galerkin method
4. The Finite Element Method (FEM)
5. Convergence and stability of the solution. Numerical issues.

6. ANALYSIS OF CONTINUA 2D

1. The Finite Element Method for continuous systems
   1. Lagrangian elements
   2. Isoparametric elements. Numerical integration
   3. Equivalent nodal forces
   4. Post-processing. Stress evaluation and recovery
   5. Error estimates and Rate of convergence
   6. Locking issues
2. Stationary problems
3. Time-dependent problems. Semidiscretization

7. FRAMES

1. Hermite shape functions. Continuity requirements.
2. General method for the calculation of the shape functions.
3. Higher order beam models.
4. Stiffness and Mass matrices
5. Equivalent nodal forces
6. Post-proecssing of the results and errors.

8. NON LINEAR ANALYSIS WITH FEM

1. Elements of incremental analysis
   1. Newton's method
   2. Inplicit and explicit methods
2. Material non linearities
   1. Fundamentals of plasticity for continuous systems. Drucker's postulate. Associated and non associated plasticity.
   2. Computatioal analysis of plastic deformation. The return mapping algoriithm.
   3. A framework for material non linear analysis of a continuous structure.
   4. Elastic-plastic beams with concentrated hinges and with diffused plasticity.
3. Geometric non linearities
   1. Geometric stiffness matrix.
   2. Linearized stability analysis.
   3. Incremental analysis and P-Delta effects.

9. PLATES

1. The equations of the elastic plate
   1. Kirchhoff-Love & Reissner-Mindlin models
   2. Generalized strains and stresses
   3. Equilibrium equations of thin plates and boundary conditions
   4. Rectangular plates with various boundary conditions
   5. Variational solutions
2. Stability of plates
   1. von Karman equations
3. Shell finite elements
   1. Degrees of freedom
   2. Interpolation of the normal
   3. 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]

6. S. Timoshenko and S.W. Krieger - Theory of plates and Shells. 2°ed Mc-Graw-Hill [TIMO]

The students are required to complete the exercises appointed during the practical lessons. It is possible to schedule an intermediate test. The final exam consists in the discussion of a written tests and in an oral colloquium.

Exercises and examples on projects of previous years are handled to the students.