1. Introduction to Advanced Structural Mechanics Lecturer: PhD Student Antonio Palermo

2. Outline of the Course Geometrical Properties of Beam Cross-Sections. Basics of Solid Mechanics: Displacements and Strains, Stress and Equilibrium, Constitutive Equations. Internal Forces in Loaded Beams: Axial Force, Bending Moment, Shear Force and Torque. The Euler-Bernoulli beam model. Analysis of Statically determinate and indeterminate Structures.

3. Tuesday 16/09/2014 9:00-13:00 Wednesday 17/09/2014 9:00-13:00 Wednesday 24/09/2014 15:00-18:00 Thursday 25/09/2014 9:00-11:00 Friday 26/09/2014 15:00-18:00 Wednesday 1/10/2014 9:30-13:00 Schedule

4. Contacts: antonio.palermo6@unibo.it Suggested reading: Beer, Johnston, DeWolf, Mechanics of Materials. Gere and Timoshenko, Mechanics of Materials. Links and resources

5. Outline of the Lecture Beam: Geometric Model. Cross-Section Geometric Properties: Area First Order Moments Centroid Second Order Moments Translation and Rotation of axes Principal axes and central Ellipse of Inertia.

6. Beam: Geometric Model A beam is a structural element generated by a planar figure Ω (i.e. cross section) that moves in the space remaing normal to the trajectory described by its centroid. Ω

7. Geometrical Requirements: Ω(s) constant or can vary continuously: BEAM YES! Ω h b l Beam: Geometric Model YES! Ω(s) NO! Ω=const

8. For each cross section Ω, it is possible to define its inertia properties that are related only to the cross section geometry. Ω Area A Static Moments Centroid G Inertia Moments Cross-section Geometric Properties G

9. Area: Cross-section Geometric Properties

10. First Moment of Area: Static Moments Cross-section Geometric Properties

11. Centroid: The centroid G of a plane figure or two- dimensional shape is the arithmetic mean position of all the points in the shape. Equivalently, the centroid G of an area is the point of intersection of all the straight lines that subdivide the plane figure in equal parts Cross-section Geometric Properties

12. Centroid: Cross-section Geometric Properties

14. Second Moment of Area: Cross-section Geometric Properties

15. Second Moment of Area: Cross Moment and Polar Moment Cross-section Geometric Properties

17. Translation of Axes: Cross-section Geometric Properties

18. Translation of Axes: Static Moment Cross-section Geometric Properties Parallel axis theorem:

19. Translation of the Axes: Second Moment of Area Parallel axis theorem: Cross-section Geometric Properties

20. Rotation of the Axes: Cross-section Geometric Properties

21. Rotation of the axes: Second Moment of Area Cross-section Geometric Properties

23. Principal axes (2/3) Cross-section Geometric Properties

24. Principal axes (3/3) : Properties If a figure has an axis of symmetry, one of the principal axis is the axis of symmetry. The other principal axis is perpendicular to the first and passes through the centroid. Cross-section Geometric Properties

25. Mohr circle: Cross-section Geometric Properties

26. Mohr circle: Cross-section Geometric Properties *A. Di Tommaso. Geometria delle Masse

27. Radius of gyration & Ellipse of Inertia Cross-section Geometric Properties

28. Radius of gyration & Ellipse of Inertia The Ellipse of Inertia provides a graphical representation of the inertia properties of the cross-section. Cross-section Geometric Properties *A. Di Tommaso. Geometria delle Masse