Lecturer: Epifanio Giovanni Virga  

Course name: Rational Mechanics
Course code: 500153
Degree course: Ingegneria Edile-Architettura
Disciplinary field of science: MAT/07
L'insegnamento costituisce attività di base per: Ingegneria Edile-Architettura
University credits: CFU 6
Course website: http://smmm.unipv.it/teaching.html

Specific course objectives

The course aims at illustrating the relevance of mathematical modeling for structural mechanics, paying special attention to equilibrium and stability.

Course programme

Tensor Algebra
Euclidean space, vectors and tensos; Transposition theorem; Symmetric and skew-symmetric tensors; Matrices representing tensors; Diadic product; Ricci's alternator; Vector product; Isomorphism between vectors and skew-symmetric tensors in three space dimensions; Spectral theorem; Orthogonal group; Adjugate of a tensor; Orientation of bases.

Differentiable curves in 3D space
Curvature and torsion; Frénet-Serret formulae.

Systems of applied vectors
Central axis; Equivalent systems.

Inertia of systems
Material symmetries; Inertia tensor; Principal directions and principal moments of inertia; Ellipsoid of inertia; Huygens-Steiner theorem; Composition theorem.

Kinematics
Observers; Changes of observer; Spin tensor and angular velocity; Poisson formula; Transformation formulae of classical relativity; Koenig theorem; Rigid body kinematics.

Dynamics
Newton's principles; Critics of Newton's principles; Galilean transformations; Relative dynamics; Force catalogue; Equilibrium of internal forces; Configurations and velocity discrete fields; Kinetic energy theorem; Energy conservation; Rigid body dynamics; Balance equations in rigid body dynamics; Euler equations for rigid body dynamics.

Lagrangian dynamics
Holonomic constraints; Lagrangian coordinates; Configuration space; Virtual motion; Non-dissipative constraints; Lagrange equations; Lagrange function.

Equilibrium theory
Balance equations; Lagrangian equilibria; Principle of virtual work; Equilibrium of rigid structures; Constraints catalogue; Hyperstatic and isostatic structures; Three-hinged arch; Internal actions in trusses; Unilateral contact problems; Equilibrium stability; Bifurcations diagrams; The theorem of Lagrange-Dirichlet; Instability criteria: two Lyapunov's theorems, Cetaev's theorem, and Hagedorn-Taliaferro theorem; Normal modes; Small oscillations.

Equilibrium of one-dimensional continua
Balance equations; Equilibrium of cables; Conservative forces; Parallel forces; Effects of friction; Equilibrium of beams; Euler elastica; Critical load.

Course entry requirements

Basic mathematics, especially Geometry and Calculus.

Course structure and teaching

Lectures (hours/year in lecture theatre): 60
Practical class (hours/year in lecture theatre): 20
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

P. Biscari, C. Poggi, E.G. Virga. Mechanics Notebook. Liguori. Napoli, 2^ edizione, 2005.

Testing and exams

Final written test, possibly followed by an oral exam (if so wishes the student).