B001521 - STRUCTURAL MECHANICS

Italian

The course presents the basic foundations of continuum mechanics; linear elasticity is also discussed in details; the students are also introduced to the equilibrium analysis of elastic framed structures.

Lecture notes.

Further suggested readings:
R. BALDACCI, Scienza delle Costruzioni, UTET
E. VIOLA, Esercizi di Scienza delle Costruzioni, vol. 1 e 2

At the end of the course the students will be able (1) to evaluate the distribution of the action characteristics over framed structures; (2) to determine the stress distribution over sections of elastic slender cylinders; (3) to evaluate the deformed configuration of framed structures made of elastic beams; (4) to perform elementary safety analyses connected with elastic instability and transition from elastic to plastic or brittle states.

It is necessary a non-trivial knowledge of the topics pertaining to the courses of Calculus 1 and 2, Geometry, Rational Mechanics.

Classroom lectures and exercises.

Written and oral exams.

1 Bodies, deformations and strain measures
1.1 Representation of bodies
1.2 Deformations
1.3 The deformation gradient F
1.4 Formal adjoint F∗ and transpose FT of F
1.5 Homogeneous deformations and rigid changes of places
1.6 Linearized rigid displacements
1.7 Kinematic constraints on rigid bodies
1.8 Kinematics of a 1D rigid body in 3D space
1.9 Kinematics of a beam system
1.10 Method of the flat link chains
1.11 Changes in volumes and the orientation-preserving property
1.12 Changes in oriented areas: the Nanson formula
1.13 Finite strain measures
1.14 Small strain measures
1.15 Finite elongation of curves and variations of angles
1.16 Deviatoric strain
1.17 Motions

2 Observers
2.1 An enlarged definition
2.2 Classes of changes in observers
2.3 Objectivity
2.4 Comments and generalizations

3 Forces, torques, balances
3.1 A preamble on the notion of force
3.2 Bulk and contact actions
3.3 Inertial effects and mass balance
3.4 Pointwise balances in Eulerian reprentation from the invariance of the external power under isometry-based changes in observers
3.5 Inner power as a derived entity
3.6 Balance equations in Lagrangian representation
3.7 Virtual work in small−strain setting
3.8 Remarks on power−invariance procedure
3.9 Discontinuity surfaces
3.9.1 Classification
3.9.2 Geometryandanalysis

4 Constitutive structures: basic aspects
4.1 Essential motivation and basic principles
4.2 Examples of material classes
4.3 A−priori constitutive restrictions and the mechanical
dissipation inequality
4.3.1 Simple bodies
4.3.2 Viscous bodies: non−conservative stress component
4.3.3 Elastic materials in isothermal setting: further constitutive restrictions
4.3.4 Constitutive restrictions for elastic materials in small strain regime
4.3.5 Linear elastic constitutive restrictions
4.4 Material isomorphisms and symmetries: isotropic materials, emergence of the pressure etc.

5 Topics in linear elasticity
5.1 Minimum of the total energy
5.2 Minimum of the complementary energy
5.3 Betti’s reciprocal theorem
5.4 Kirchhoff’s theorem
5.5 Navier’s equations and the bi-harmonic problem

6 The de Saint-Venant problem
6.1 Statement of the problem
6.1.1 Geometry
6.1.2 Loads
6.1.3 Material
6.1.4 Ansatz on the structure of the stress
6.2 Global balances for a portion of the cylinder
6.3 Explicit expression for 
6.3.1 c = 0
6.3.2 The Navier polynomial
6.4 The neutral axis
6.5 A scheme for evaluating 
6.6 Approximate evaluation fo the stress due to shear: the Jourawski formula
6.7 Shear actions may produce torsion effects: the shear centre
6.8 Preliminaries to the analysis of the torsion
6.8.1 Further remarks on the shear stress potential function over the cross-section
6.8.2 The Prandtl function
6.9 The torsion moment
6.10 Torsional curvature
6.11 Maximality of the shear stress due to torsion
6.12 Displacements due to torsion: warping
6.13 Analysis of cross-sections with multiple connection under torsion
6.14 The Toupin theorem: proof of the de Saint-Venant principle

7 Yield criteria
7.1 Reasons for introducing yield criteria
7.2 Some prominent criteria
7.2.1 The Tresca criterion
7.2.2 The Beltrami criterion
7.2.3 The Huber-von Mises-Hencky criterion
7.2.4 The Drucker-Pragher criterion
7.2.5 The Hill criterion

8 Elastic framed structures
8.1 The elastica
8.2 The Timoshenko beam
8.3 Principle of virtual power for elastic beams: the force method
8.4 Hyperstatic structures

9 Elastic stability
9.1 Qualitative characterization of the equilibrium
9.2 The Euler beam