Introduction to Advanced Structural Mechanics
Lecturer: PhD Student Antonio Palermo
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.
• 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
• Contacts:[email protected]
Suggested reading:
• Beer, Johnston, DeWolf, Mechanics of Materials.• Gere and Timoshenko, Mechanics of Materials.
Links and resources
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.
Beam: Geometric ModelA 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
Geometrical Requirements:• Ω(s) constant or can vary continuously:
BEAM
YES!
Ω
h
bl
h≅ 𝑏<5 𝑙
Beam: Geometric Model
YES!Ω(s)
NO!
Ω=const
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
Area:
:
Cross-section Geometric Properties
𝑑𝐴=𝑅𝑑𝜃 𝑑𝑟
First Moment of Area: Static Moments
Cross-section Geometric Properties
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
Centroid:
Cross-section Geometric Properties
Static Moment & Centroid: Properties
• either • The Static Moment calculated with respect to an axis of
symmetry = 0.• If an area has an axis of symmetry, the centroid G lays on the
axis.• If an area has two axes of symmetry, the centroid G is located in
the intersection of the axes .• (Domain of Integration can be added)
Cross-section Geometric Properties
Second Moment of Area:
Cross-section Geometric Properties
Second Moment of Area: Cross Moment and Polar Moment Cross Moment
Polar Moment
Cross-section Geometric Properties
Second Moment of Area: Properties
•
• when P is the origin of the x,y axes.
• if x or y are axes of symmetry • (valid for all the Second Moment of Area)
Cross-section Geometric Properties
Translation of Axes:
Cross-section Geometric Properties
Translation of Axes: Static Moment
Cross-section Geometric Properties
Parallel axis theorem:
Translation of the Axes: Second Moment of Area
Parallel axis theorem:
Cross-section Geometric Properties
Rotation of the Axes:
Cross-section Geometric Properties
Rotation of the axes: Second Moment of Area
Cross-section Geometric Properties
Principal axes (1/3)Idea: find the for whichand are the maximum and minimum moment of inertia (or viceversa)
with
Cross-section Geometric Properties
Principal axes (2/3)With we define , principal axes with: • principal moment of Inertia
(minimum/maximum moment of Inertia or viceversa )
with:
and:
Cross-section Geometric Properties
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
Mohr circle:Given the principal axes , with
Parametric equations of a circle in the plane ,
Cross-section Geometric Properties
Mohr circle:Parametric equations of a circle in the plane , Cross-section Geometric Properties
*A. Di Tommaso. Geometria delle Masse
Radius of gyration & Ellipse of Inertia
Analytical Expression:
Cross-section Geometric Properties
Radius of gyration & Ellipse of InertiaThe 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
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