Università Roma Tre

Kinematics of rigid bodies. The rigid body model. Rigid displacements. General formula of infinitesimal rigid displacement. Scalar representation of the rigid displacement field. Planar rigid displacements. Systems of rigid bodies.

Kinematic characterization of a constraint. External constraints and internal constraints.
The cinematic problem. Kinematic classification.

Statics of rigid bodies. External forces. Force, moment of a force, systems of forces, force density, distributed loads.
Static characterization of constraints. The static problem. Cardinal equations of statics. Static classification. Statical-kinematical duality.

Beam kinematics. Displacement, rotation, hypothesis of small displacements. Kinematic conditions.

Deformation measurements in the Timoshenko model. Axial deformation. Angular scroll. Bowing. Equations of congruence. Model of Euler-Bernoulli. Vector representation of congruence equations. The kinematic problem for the beam. Statics of the beam. External actions and internal actions. Partwise balance. Differential equilibrium equations in scalar and vector format. The static problem. Internal action diagrams.

Constitutive equations. Phenomenology of the response of a material. The uniaxial test. Elastic behavior. Plastic behavior and rupture. Ductile and fragile materials.

Constitutive equations for the elastic beam. Axial behavior, flexural behavior, shear behavior. Uniform thermal variation, affine thermal variation.

The elastic problem for the beam and its formulation.

Displacement method. Equation of the tension beam. Inflexion of a beam in the Euler-Bernoulli model. Extension to the Timoshenko model. Connection conditions and problem formulation for beam systems. Kinematic and static performance of internal constraints.

Principle of virtual work. The concept of congruent system. Notion of a balanced system. External virtual work. Internal virtual work. Theorem of virtual works, statement and proof. Application of the Virtual Works Principle to the calculation of displacements and rotations in statically-determined structures.

Force method. Released structure. Application of the method to the general case. Müller-Breslau equations. Flexibility matrix. Effect of constraint displacement and thermal distortions.

Reticular trusses. Internal isostaticity of the triangular mesh. Canonical nodes. Node method. Canonical sections. Ritter method.

Continuous beams. The three-moment equation.

Three-dimensional continuous bodies: analysis of the deformation. The strain tensor. Geometrical interpretation of the components of the strain tensor.
State of triaxial deformation. State of cylindrical deformation. Spherical or hydrostatic deformation state. Mohr's circles.

Three-dimensional continuous bodies. Cauchy's concept of traction. Partwise balance. Cauchy's Lemma. The stress tensor. Differential equations of equilibrium.

Principal stresses and principal directions. Voltage states. Lamé's ellipsoid of tension. Isostatic lines. State of plane or biaxial tension. Purely tangential tension state. Uniaxial voltage state. Mohr's circles for the stress.

Linearly elastic constitutive equations. Experimental determination of elastic constants. Traction test. Torsion test. Isotropic materials: the generalized Hooke law.

The problem of elastic balance. The Virtual Works Theorem. Partial solutions to the problem of elastic equilibrium. Deformation work. Clapeyron's theorem. Betti's theorem. Theorem of the minimum total potential energy. Theorem of the minimum total complementary potential energy

The problem of Saint Venant. Postulate of Saint Venant. The semi-inverse method.

Normal force. Flexure.

Torsion of compact and hollow circular sections.

Torsion of compact sections of arbitrary shape. The Neumann problem. Elliptical sections. Polygonal sections. The hydrodynamical analogy. Thin rectangular section. Open sections composed of thin rectangles. Thin-walled sections: Bredt's theory.

Bending and shearing. Distribution of normal tractions. Distribution of tangential tractions: Jourawsky formula and its applicability.
Thin sections open. Thin rectangular section. Thin double T section. U and H shaped sections. Thin sections closed. Symmetrical closed section. Symmetrical compact sections. Determination of the shear center.

Rupture criteria for fragile materials and ductile materials.

The phenomenon of structural instability. Stability analysis in rigid beams with elastic restraints. Stable branches. Unstable branches. Sensitivity to initial imperfections. Euler's Elastica. Dependence of the critical load on the constraint conditions. Inflection plans. Curves of stability, slenderness.

The beam: structural analysis and verification. Extension of the Saint Venant theory. Resistance criteria for the Saint Venant solid.

P. Casini & M. Vasta, "Scienza delle costruzioni", Città Studi Edizioni 2016.
Steen Krenk & Jan Høgsberg, "Statics and Mechanics of Structures", Springer 2013.