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

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

mailto:[email protected]

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:

:


𝑑𝐴=𝑅𝑑𝜃 𝑑𝑟

First Moment of Area: Static Moments


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


Centroid:


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)


Second Moment of Area:


Second Moment of Area: Cross Moment and Polar Moment Cross Moment

Polar Moment


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)


Translation of Axes:


Translation of Axes: Static Moment


Parallel axis theorem:

Translation of the Axes: Second Moment of Area

Parallel axis theorem:


Rotation of the Axes:


Rotation of the axes: Second Moment of Area


Principal axes (1/3)Idea: find the for whichand are the maximum and minimum moment of inertia (or viceversa)

with


Principal axes (2/3)With we define , principal axes with: • principal moment of Inertia

(minimum/maximum moment of Inertia or viceversa )

with:

and:


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.


Mohr circle:Given the principal axes , with

Parametric equations of a circle in the plane ,


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:


Radius of gyration & Ellipse of InertiaThe Ellipse of Inertia provides a graphical representation of the inertia properties of the cross-section.


*A. Di Tommaso. Geometria delle Masse

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

Documents

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