Kinematic analysis of planar beam systems. Degrees of freedom and virtual displacements. The constraints.
Iperstatic, isostatic and ipostatic beam systems.
Determination of reaction forces and internal actions (axial force, shear, and bending moment) in statically determinate planar beam systems. Anomalous cases.
Planar system of trusses statically determined
THE CAUCHY'S CONTINUUM
Description of the problem.
The postulate of the additional constraints.
STRESS ANALYSIS
Cauchy's definition of tensions and stress
The Cauchy's theorem
Reference system rotation: rotation of a vector and of a tensor.
Stress components in different reference system.
Principal stresses, principal directions, and invariants
Decomposition of stresses in volumetric and deviatoric part
Plane stress. The Mohr's Circle for plane stress
STRAIN ANALYSIS
Small strain kinematics in a continuum medium: the displacement gradient tensor.
Rigid body motion and deformations.
The equations of external compatibility
Physical interpretation of the components of the tensor of small strains
Principal strains, principal strain directions and invariants of strain.
The equations of internal compatibility
PRINCIPLE OF VIRTUAL WORKS FOR CONTINUUM MEDIUMS
ELASTICITY
Uniaxial tensile test
Definition of Grenn's elasticity
The linear elastic homogeneous and isotropic material
The inverse relations
Physical meaning of the elastic constants: Young's modulus, Poisson's coefficient, shear modulus and bulk modulus. Thermodynamic limits of the elastic constants.
THE LINEAR ELASTIC PROBLEM
The principle of the effects superposition
Uniqueness and existence of the solution: the Kirchhoff Theorem.
THE SAINT VENANTS' PROBLEM
General formulation.
The Saint Venants' postulate
BENDING
The solution. Infinitesimal virtual work in bending
Tension and compression.
Structural constitutive relations for axial loads.
Neutral axis.
Right bending.
General bending
Bending and tension/compression. Structural constitutive relations for bending.
Bending and tension or compression.
The central intertia core for polygonal sections.
TORSION
Torsion in cylindrical circular beams
General solution.
Hydrodynamic analogy
Torsion in small thickness beams with closed hollow sections.
Torsion in beams with rectangular section.
Torsion in small thickness open beam
SHEAR
Approximate shear stress evaluation: computations of the average shear stress along a generic segment
Bending and shear symmetric in beams with shear parallel to the symmetry axis
Bending and shear symmetric in small thickness open beam.
The center of shear
The equation of the elastic line
Computation of displacements and hyperstatics in beams.
Non elastic effects
Elastic boundary conditions
Applications of the Principle of Virtual Works for planar beams
Computation of displacements
Computation of hyperstatics
Strength criteria
Von Mises and Tresca Criterion for ductile materials.
The Galileo criterion for brittle materials.
Instability
Computation of the Euler collapse load.