Teoria delle Strutture
Theory of Structures Lando Mentrasti
Fundamentals of theoretical and applied structural mechanics.
KNOWLEDGE AND UNDERSTANDING:
The course presents the Finite Element Method (FEM) to approximate and discretize an elastic structure that cannot be reduced to elementary models. The focus is both on the mechanical assumptions and on the reliability of the numerical result obtainable. The course is divided into two parts: Part I Matrix analysis of elastic structures. Kinematics and statics (local and global), dof control, duality, assembling, generalized constraints and conditioning. Part II Finite Element Method (FEM). Weak form of the elastic problem via the Minimum of the Potential Energy theorem: general expression of the stiffness matrix. Form functions. Main element typology. Compatibility problems. CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
The student is able to comprehend the principles and the restrictions of a FEM code and to critically assess the reliability of a numerical structural model. TRANSVERSAL SKILLS:
The presented theory and the discussed issues are the basic theoretical tools to face in autonomy and in a critical way every non-elementary structural design.
The course presents the Finite Element Method (FEM) as a tool to analyse an elastic structure not reducible to elementary models. The focus is both on the mechanical assumptions and on the reliability of the numerical result obtainable.
RIGID BODY structures (hints). Kinematics and statics matrix formulation. Principle of the virtual work of the constraint reactions. Virtual Work Theorem, duality.
MATRIX ANALYSIS of ELASTIC STRUCTURES. Local stiffness (notable properties), degree of freedom transformation, constraints and kinematical control matrix. External constraints: general treatment, static condensation, mixed variable formulation, releases, non-local constraints (identification). Global stiffness matrix: assembly, topology, bandwidth and profile. Conditioning of the algebraic solution.
FINITE ELEMENT METHOD: weak form of the elastic problem (minimum potential energy approach): regularity, discretization, locality. Shape functions: interpolation, completeness, inter-element continuity. FEM morphology: quadrilateral, rectangular, Lagrangean, Serendipity, transition elements, triangular. Derivation of the element stiffness and mass matrices. Placement of distorted element (isoparamentric element properties).
Hints on the Weak Formulation of the continuum structural problems (physical meaning).
Geometrical, non-linear and dynamics aspects behavior are occasionally sketched.
Brief history of the matrix analysis of structures.
Training in FEM, numerical and symbolic codes.
Development of the examination
LEARNING EVALUATION METHODS
Oral exposition on the main subjects presented in the lectures (with discussion of a possible numerical application, suggested by the student)
LEARNING EVALUATION CRITERIA
Suitability of the argumentation, from both the theoretical and application pont of view.
LEARNING MEASUREMENT CRITERIA
Correspondence with the arguments presented in the lectures.
FINAL MARK ALLOCATION CRITERIA
Weighted evaluation, with special regards to synthesis, distinctive aspects and completeness. Personal research and evaluation are favorably encouraged.
Corradi Dell'Acqua, Meccanica delle Strutture Vol 1 e 2, McGraw-Hill 2010
Fish, Belytschko, A first Course in Finite Elements, Wiley 2007
Luongo, Paolone, Meccanica delle Strutture, Casa Editrice Ambrosiana, 1997
Supplementary material is distribuited during the lectures and is available on the UnivPM site (or sent via e-mail on request).
- Ingegneria Civile (Corso di Laurea Magistrale (DM 270/04))