STATIC OF RIGID BODY SYSTEMS
Forces and couples. Resultant and resultant moment of a system of applied forces. Equivalent systems of forces. Elementary equivalence operations. Balance of a rigid body or a system of rigid bodies. Cardinal equations of statics. In-plane force systems. Distributed forces over a volume (gravitational force), over a surface, over a line; concentrated forces and couples. Constraint reactions; static characterization of external and internal in-plane constraints. Solution of the equilibrium problem. Isostatic, hyperstatic, labile, degenerate structures. Characteristics of internal stress in beam systems: normal stress, shear stress, bending moment, torque; indefinite equilibrium equations for the rectilinear beam.

MASS GEOMETRY
Area, static moment, center of gravity, moment of inertia, radius of gyration, mixed moment of inertia, Huygens theorem. Principal axes of inertia, central ellipse of inertia.

INTRODUCTION TO THE THEORY OF ELASTIC STRUCTURES
Limits of the rigid body model. Elementary deformable model: rod, linear elastic bond. Equilibrium, compatibility, and constitutive equations for the straight rod. The linear elastic problem; solution methods: force method and displacement method for the straight rod.

ELASTIC BEAM AND HYPERSTATIC SYSTEMS OF BEAMS
Differential relations between transverse displacement of the axis line, rotation of the straight section, and bending curvature; curvature due to thermal distortions or bending moment; integration of the elastic line equation. Deformation characteristics (bending and torsional curvature, extension, sliding); elastic relationship between stress and deformation characteristics. Virtual Work Theorem for deformable beams; application of the Theorem of Least Work to determine displacements and rotations in isostatic structures. Resolution of hyperstatic structures through compatibility equations (Muller-Breslau equations).

CONTINUUM MECHANICS. STRESS ANALYSIS
Cauchy's stress tensor. Cauchy's lemma. Decomposition of the Cauchy stress tensor. Cauchy's stress formula. Equilibrium equations. Principal stresses and directions. Triaxial, cylindrical, and spherical stress states. Stress transformation. Octahedral stress. Mohr's circles. Plane stress, pure shear, and uniaxial stress states. Mohr's circles for stress analysis at a point on the De Saint Venant beam. Isostatic lines.

DEFORMATION ANALYSIS
Rigid body motion. Decomposition of displacement into deformation tensor and rigid rotation tensor. Mechanical interpretation of deformation components: stretches and angular displacements. Decomposition of deformation. Volume change. Cauchy's strain formula. Principal strains and directions. Triaxial, cylindrical, and spherical strain states. Volume change. Mohr's circles for strain analysis. Plane strain, pure shear, and uniaxial strain states.

HOOKE'S LAW
Hooke's law for uniaxial tension, linearity, and plastic behavior. Tensile and torsion tests. Elastic behavior under triaxial stress: generalized Hooke's law. Elastic constants: Young's modulus, Poisson's ratio, bulk modulus. Plastic deformation.

THE ELASTIC PROBLEM
Existence and uniqueness of the solution to the problem of elastic equilibrium: Navier and Beltrami equations. Solution of the De Saint Venant problem for bending, tension, and torsion. Neumann problem for torsion.

THEORY OF ELASTIC BEAMS
Determination of stress and deformation for a linear elastic material beam starting from stress characteristics: conservation of flat sections; normal stress centroid; pure bending; curved bending. Twist in closed thin sections; Bredt's formula. Twist in elongated rectangular sections; C, L sections. Torsion in circular sections. Approximate shear stress calculation (Jourawski). Combined stresses: deflected bending; eccentricity of normal stress; cutting and torsion.

FAILURE CRITERIA
Failure criteria in monoaxial and triaxial stress states. Criteria for brittle materials: Galileo-Rankine, De Saint Venant-Grashof. Criteria for ductile materials: Tresca, von Mises-Hencky.

STRUCTURAL INSTABILITY
Stability analysis of rigid beams with elastic constraints. Euler's column, stability curves, slenderness.