Università Roma Tre

Vectors. Moment of a vector with respect to a point. Resultant force and resultant moment of a system of vectors. Equivalence of systems of vectors. Planar systems. Linear transformations on vector spaces. Second-order tensors. Differentiation of vector and tensor functions. Gradient and divergence. Divergence theorem. Stokes’ theorem.
Centroid of a material system. First moments and moments of inertia of a planar region. Moments of inertia of a planar region with respect to lines passing through a given point. Mohr’s circles. Polarity and involutions determined by the ellipse of inertia.
Equilibrium of a material point. Constraints. Infinitesimal rigid displacements. Euler’s theorem. Work in a rigid displacement. Equilibrium equations of rigid bodies. Beams and frames. Diagrams of the components of force and couple resultants. Forces exerted by the constraints. Principle of virtual work for rigid bodies. Statically determined structures. Graphic statics.
Deformation of a continuous body. Measures of deformation. Principal strains. Volume change. Compatibility conditions. Equilibrium equations of deformable bodies. Stress tensor. Principal stresses. Plane stress. Moor’s circles for stresses. Constitutive equations. Hyperelastic materials. Energy functional. Clapeyron’s theorem. Equilibrium boundary-value problems. Betti’s theorem. Virtual work principle for deformable bodies. Von Mises and Tresca yield criteria.
Saint-Venant’s problem. Extension, bending, torsion, and shear of beams. Equilibrium differential equations for a beam. Limitations of the theory. Allowable stress. Euler’s critical load.
Effects of imperfect constraints and temperature variations. Mohr’s analogue. Hyperstatic beams. Continuous beams. Analysis of planar trusses and frames by means of virtual work principle, force and displacement methods.

Notes written by the teacher. Learning material, as notes, slides employed for the lectures, and applicative examples, is available online on the moodle page of the course.