Lecturer: Epifanio Giovanni Virga  

Course name: Analytical mechanics
Course code: 500397
Degree course: Ingegneria Edile-Architettura
Disciplinary field of science: MAT/07
L'insegnamento costituisce attività di base per: Ingegneria Edile-Architettura
University credits: ECTS 6
Course website: n.d.

Specific course objectives

Course programme

Methodological Introduction
Euclidean space; Translation space; Points and vectors; Geodesics on surfaces; Inner product; Euclidean distance; Euclidean isometries.

Tensor Algebra
Tensors; Tensor product of vectors; Orthogonal projections; Transpose of a tensor; Tensor space; Symmetric tensors; Shew-symmetric tensors; Additive decomposition of a tensor; Eigenvalues and eigenvectors of a tensor; Components of a vector; Matrix representation of a tensor; Spectral theorem; Diagonal representation of a symmetric tensor; Ricci’s alternator; Representation of the cross product of vectors; Skew-symmetric tensors; Axial vector associated with a skew-symmetric tensor; Matrix representation of a skew-symmetric tensor; Orthogonal tensors; Rotations and reflections.

Inertia
Center of mass; Axial moment of inertia; Tensor of inertia; Principal basis of inertia; Principal moments of inertia; Plane systems; Cullmann ellipse; Material symmetries; Axis of mirror symmetry; Center of mass and tensor of inertia for symmetric beam cross-sections; Composition theorem.

Differential Properties of Curves in Space
Arc-length parameter; Unit tangent vector; Curvature and torsion; Frenet-Serret equations; Frenet-Serret frame; Osculating, normal, and rectifying planes; Osculating circle; Radius of curvature; Sign of the torsion; Cartesian formulas for curvature and torsion; Frenet-Serret twist vector; Cosserat frame; Cosserat twist vector; Cosserat strains; Constrained Cosserat frames.

Balance Equations
Stress principle; Resultant stress vector; Resultant couple; External force and couple densities; Integral balance equations of forces and moments; Local form of the balance equations.

Kirchhoff Beam
Kirchhoff hypotheses and their translation in the Cosserat formalism; Bending (flexural) rigidities; Twisting (torsional) rigidity; Planar, untwisted beams.

Euler Beam
Equilibrium equation and boundary value problem; Euler’s critical load; Bifurcation problem.

Singular Loads
Transmission conditions; Beam subject to concentrated forces and couples; Tie beam; A posteriori estimate of the error in the small deflection approximation for the tie beam.

Cable Theory
Equilibrium equation: balance of forces; Tension; Indefinite equation and boundary conditions; Intrinsic form (in the Frenet-Serret frame); Equilibrium of cables under the action of a system of parallel forces. Resolvent equation for the Cartesian representation of the equilibrium shape; Law of tension.

Catenary Theory
Determination of the equilibrium shape; Law of tension; Minimum tension; Maximum tension; Tension optimization: minimax problem; Catenary with a concentrated weight.

Suspension Bridge
Shape of the suspension cable; Maximum and minimum tension: monotonicity in the cable’s length; Cable subject to a concentrated load: transmission condition.

Course entry requirements

Course structure and teaching

Lectures (hours/year in lecture theatre): 80
Practical class (hours/year in lecture theatre): 0
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

P. Biscari, C. Poggi, E.G. Virga. Mechanics Notebook. Liguori, 2005.

Titolo del riferimento da modificare.

Testing and exams

Final written test.