AFTER A PRESENTATION OF THE FUNDAMENTAL CONCEPTS ON DEFORMATION AND EQUILIBRIUM OF CONTINUOUS BODIES IN LINEAR ELASTICITY, THE COURSE FOCUSES ON SAINT-VENANT’S BEAM THEORY AND ITS APPLICATIONS TO THE ANALYSIS OF TYPICAL STRUCTURES OF CIVIL ENGINEERING. AT THE END OF THE COURSE, A STUDENT SHOULD BE ABLE TO DETERMINE THE STATE OF DEFORMATION AND STRESS IN SIMPLE STRUCTURES AND TO EVALUATE THEIR SAFETY UNDER THE ACTION OF GIVEN LOADS.
teacher profile teaching materials
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 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 exerted by the constraints. Principle of virtual work for rigid bodies. Statically determined structures. Graphic statics.
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. Moor’s circles for stresses. Constitutive equations. Hyperelastic materials. Energy functional. Clapeyron’s theorem. Equilibrium boundary-value problems. Betti’s theorem. Virtual work principle for deformable bodies. Von Mises and Tresca yield criteria.
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.
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
Programme
Vectors. 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.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 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 exerted by the constraints. Principle of virtual work for rigid bodies. Statically determined structures. Graphic statics.
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. Moor’s circles for stresses. Constitutive equations. Hyperelastic materials. Energy functional. Clapeyron’s theorem. Equilibrium boundary-value problems. Betti’s theorem. Virtual work principle for deformable bodies. Von Mises and Tresca yield criteria.
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.
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
Core Documentation
Notes written by the teacher, published on the “moodle” page of the course. Reference texts are given in the bibliography of the notes.Type of evaluation
The final evaluation of the students is made by means of a written test, usually regarding applications, and an oral examination, more oriented towards the theoretical fundaments.