Università Roma Tre

Vector Theory. Vector quantities. 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.
Geometry of material systems. 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 rigid bodies. 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 exercised by the constraints. Principle of virtual work for rigid bodies. Statically determined structures. Graphic statics.
Equilibrium of deformable bodies. 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. Mohr’s circles. Constitutive equations. Hyperelastic materials. Clapeyron’s theorem. Equilibrium boundary-value problems. Betti’s theorem. Virtual work principle for deformable bodies. Von Mises and Tresca yield criteria.
Beam theory. 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.
Theory of structures. 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.
Introduction to plastic analysis of structures. Plastic moment of a beam section. Plastic hinges. Collapse of a structure due to transformation into a mechanism. Load factor. Static and kinematic theorems.

Notes written by the teacher, published on the “moodle” page of the course. Reference texts are given in the bibliography of the notes.