2E4: SOLIDS & STRUCTURES

Lecture 8

Dr. Bidisha GhoshNotes: http://www.tcd.ie/civileng/Staff/Bidisha.Ghosh/Solids & Structures

Properties of Sections

Cross-sections of beams:

Cross-Sections of other structural or machine elements:

To find out stress or deformation we need to know about the geometric properties of these sections!

Centroid

Centroid is the geometric centre which represents a point in the plane about which the area of the cross-section is equally distributed.

Centre of gravity for a body is a point which locates the gravity or weight of the body.

Centroid and CG are same for homogeneous material.

Moment of Area

dA

A

dA

This is called the First Moment of AreaAn important concept to find out centroid.

• The limits of the integration are decided based on the dimensions (end points) of the area under consideration.

x y

A A

Q ydA Q xdA

•Take a infinitesimally small area (dA) in the shaded area (area under consideration).

•Moment of this area about the point O,

•Moment of the entire shaded area about the point O can be by summing over all such small dA areas or by,

•First moment of area about x-axis or y-axis,

Calculating position of centroid

The centroid of the entire shaded area (set of areas dA) is the point C with respect to which the sum of the first moments of the dA areas is equal to zero.

The centroid is the point defining the geometric center of system or of an object.

yx

A A

QQx ydA y xdA

A A

Centroid of a Triangle

Composite areas

When a composite area is considered as an assemblage of n elementary areas, the resultant moment about any axis is the algebraic sum of the moments of the component areas.

Therefore the centroid of a composite area is located by,

i i i i

i i

A x A yx y

A A

Centroid of an L-Shaped Area

Centroid of an L-Shaped Area

Moments of InertiaMOI is a measure of the resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics.

It is the second moment of area,

Radius of gyration, (the distance at which the entire area can be assumed to be distributed for calculation of MOI)

yxx y

IIr r

A A

2 2 x y

A A

I y dA I x dA

Can you write a matlab/excel code

to calculate moment of inertia?

Moments of Inertia of a rectangle

2 2

2 22

2 2

22

2

3 32

2

=

=3 12

x AA

d b

d b

d

d

d

d

I y dA y dxdy

y dx dy

y bdy

y bdb

Polar Moment of Inertia

This is the moment of inertia of a plane area about an axis perpendicular to the area.

0 x yJ I I

Parallel Axis Theorem

The parallel-axis theorem relates the moment of inertia of an area with respect to any axis to the moment of inertia around a parallel axis through the centroid.

2 2( ) = x x y

A

I y y dA I Ad

Moment of Inertia of an I-beam