The course aims to teach to the students the basic concepts of the mechanics of materials and of the mechanics of soids, with special attention to the behaviour of mevhanical structures. Main topics of the course are the principles of equilibrium, the mechanics of deformable continua, the theory of elasticity, the analysis and design of the performance of structural elements respect to resistance and deformation. The topics of the course are typically those covered in the courses of Statics and Strength of Materials.
The couse is subdivised in four parts:
The first part: analysis of the discrete structural systems.
The second part: Analysis of the stress and strain for a Cahuchy continua.
The third part: Linear-elastic constitutive behaviour.
The fourth part: Position of De Saint Venant problem.
1. ANALYSIS OF THE MECHANICAL SYSTEM Modeling. Selection procedure, description and idealization of the structural scheme, loads and constraint definitions. Summary of the methods of statics. Analysis of the structural systems: loads and reactions. |
[BJ] Cap.1 [CC]Cap.1.1 |
2. CONSTRAINTS constraint definitions. Classification of the constraints for the 2D and the 3D cases. Free Body Diagram. |
[CC] Cap.1.2 |
3. EQUILIBRIUM OF DISCRETE SYSTEMS Equilibrium equations of structures composed by a finite number of elements. Equilibrium of material points and systems of cables. Equilibrium equations of articulated systems and beam systems. Auxiliary equilibrium equations. |
[BJ] Cap.1 [CC] Cap.1.2 |
4. BEAM MODEL Stress resultants and differential equations of the equilibrium. Definition of the internal stress resultant distributions. 2D and 3D systems composed by beams. Design of beam loaded by axial and bending force. |
[CC] Cap.1.3 |
5. STRESS AND STRAIN ANALYSIS Normal and tangential componenet of the stress. Stress Tensor definition and transformation of the stress components for the plane tension state. Principal Stresses, Mohr circle. Chauchy equilibrium equations. Definition of the strain tensor. Small strains. Axial and shear strain components. Principalstrains, Morh circle. Compatibility equations. |
[BJ] Cap.1, 2 e 7 |
6. PRINCIPLE OF THE VIRTUAL WORK Principle of the virtual work for deformable continuum and its particularization to the structural models |
[BJ] Cap.11 e 12 |
7.CONSTITUTIVE EQUATION Hooke model and its generalization to the 3D case. Elastic deformation energy. Elasto-plastic behaviour. Yelding criteria (Tresca and Von-Mises Criteria). |
[BJ] Cap.2 |
8. AXIAL FORCE De Saint Venant's principle. Position of the De Saint Venant problem. Definition of stress and strain for beam under axial force. Composite structures and static indeterminated systems. Plastic axial force. |
[BJ] Cap.2 |
9. TORSION Torsion deformation of beam with circular cross sections. Definition of the stress and strain for the torsion. Composite structures and static indeterminated systems. Elasto-plastic torsion. Stress in non-circular elements.Warping function and generalization of the torsion. Hollow shafts with thin wall. |
[BJ] Cap.3 |
10. BENDING Stress and strain of beam under bending deformation. Symmetric membrer in pure bending. Pure bending and unsymmetric bending. Elasto-plastic bending for symmetric members. Eccentric axail force. Analysis and design of beam for bending. |
[BJ] Cap.4 |
11. SHEAR (NON UNIFORM BENDIG) Shearing Stresses in beam for the compact cross section case and the thin-walled case. Unsymmetric loading of thin walled members. Shear Center. |
BJ] Cap.6 |
12. VERIFICHE DI RESISTENZA Definition of the safety factor for beam. |
[BJ] Cap.5 e 8 |
13. ELASTIC DEFLECTION OF BEAM Definition of the pure bending deformation of a beam. Equation for the elastica. Method of the unitary force or application of the principle of the virtual work. Solution of statically indeterminate system composed by beams. |
[BJ] Cap.9 |
14. STABILITY OF STRUCTURES Euler's Formula for pin-ended columns. Extension of the Euler's formula to columns with other end conditions. Design of compressed members. |
[BJ] Cap.10 |
1. [BJ]
Meccanica dei Solidi, 5° ed., F. P. Beer, E. R. Johnston, D.F. Mazurek , S. Sanghi, Mc Graw Hill
2. [CC]
Lezioni di Scienza delle Costruzioni, L. Gambartta, L. Nunziante, A. Tralli