STATIC OF RIGID BODY SYSTEMS. Forces and pairs. 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. Flat force systems. Forces spread over a volume (gravitational force), over a surface, over a line; concentrated forces and couples. Binding reactions for smooth, fixed and bilateral constraints; static characterization of external and internal plan constraints. Search for balanced reactive states. Isostatic, hyperstatic, labile, degenerate structures. Characteristics of internal stress in beam systems: normal stress, shear stress, bending moment, torque; indefinite equations of equilibrium for the rectilinear beam.
MASS GEOMETRY.
Area, static moment, center of gravity, moment of inertia, radius of inertia, mixed moment of inertia, Huygens theorem. Main 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, congruence and constituent 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 bond between the characteristics of stress and deformation. Virtual Works Theorem for the deformable beam; application of the TLV to search for displacements and rotations in isostatic structures. Resolution of hyperstatic structures through congruence equations (Muller-Breslau equations).
CONTINUOUS OF CAUCHY. STRESS ANALYSIS . Cauchy's tension. Cauchy lemma. Decomposition of the Cauchy stress vector. Cauchy formula. Indefinite equilibrium equations. Main stress and directions around the point. Triaxial, cylindrical and spherical tension states. stress switch. Octahedral stress. Main reference. Circumferences of Mohr. State of plane tension, purely tangential and uniaxial. Mohr circles for stress analysis at a point on the De Saint Venant beam. Isostatic lines.
DEFORMATION ANALYSIS. Act of rigid motion. Decomposition of the displacement around the point: deformation tensor and rigid rotation tensor. Mechanical meaning of deformation components: elongations and angular displacements. Decomposition of the deformation process. Cubic dilatation. Cauchy's formula for deformation. Main directions of deformation. Triaxial, cylindrical and spherical deformation states. Cubic dilation. Main reference. Mohr circumferences for deformation analysis. State of plane deformation, pure sliding and uniaxial.
HOOKE'S LAW. Hooke's law for the state of uniaxial tension, linearity and plastic behavior. Tensile test and torsion test. The elastic bond in the triaxial regime: generalized Hooke's law. Elastic constants: Young's modulus and Poisson's ratio, cubic modulus of elasticity. Inelastic deformations.
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 pressure bending, tension approach. The Neumann problem for torsion, approach to displacements.
THE SOLID OF DE SAINT VENANT. Determination of the stress and deformation state for a homogeneous linear elastic material beam starting from the characteristics of the stress (technical treatment): hypothesis of conservation of the flat sections; normal effort centered; straight bending; near straight and deviated bending. Twist in the closed thin sections; Bredt's formula. Twisting in elongated rectangular sections; C, L sections or in any case developed in a thin rectangle. Torsion in the full circular section. Approximate treatment of Taglio (Jourawski). Compound stresses: deflected bending; eccentric normal stress, relation between stress center and neutral axis, central inertial core; cutting and twisting, cutting center.
RESISTANCE CRITERIA. Resistance criteria in the monoaxial and triaxial regime. Resistance criteria for fragile materials: Galileo-Rankine, De Saint Venant-Grashof. Resistance criteria for ductile materials: Tresca, Octahedral tension (Huber-Mises-Henchy).
APPROXIMATE METHODS: THE FINITE ELEMENT METHOD. Outline of approximate methods for the evaluation of elastic structural response. The theorem of the minimum total potential energy (statement). The Rayleigh-Ritz method. The finite element method: finite element rod, finite element beam. Application examples.
