Moment of Area

7/27/2019 Moment of Area

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2E4: SOLIDS & STRUCTURES

Lecture 8

Dr. Bidisha Ghosh

Notes:

http://www.tcd.ie/civileng/Staff/Bidisha.Ghosh/

Solids & Structures


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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!


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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.


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Moment of Area

dA

A

dA

This is called the First Moment of Area

An 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 underconsideration).

Moment of this area about the point O,

Moment of the entire shaded areaabout the point O can be by summing

over all such small dA areas or by,

First moment of area about x-axis ory-axis,


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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 definingthe geometric center ofsystem or of an object.

yx

A A

QQx ydA y xdA

A A


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Centroid of a Triangle


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Composite areas

When a composite area is considered as an

assemblage of nelementary areas, the resultantmoment 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 yA A


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Centroid of an L-Shaped Area


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Centroid of an L-Shaped Area


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Moments of Inertia

MOI is a measure of the resistance to changes to itsrotation. It is the inertia of a rotating body with respect to its

rotation. The moment of inertia plays much the same rolein rotational dynamics as mass does in linear dynamics.

It is the second moment of area,

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

yxx y

IIr rA A

2 2

x yA A

I y dA I x dA

Can you write a

matlab/excel code to

calculate moment of

inertia?


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Moments of Inertia of a rectangle

2 2

2 22

2 2

22

2

3 32

2

=

=3 12

xA

A

d b

d b

d

d

d

d

I y dA y dxdy

y dx dy

y bdy

y bdb


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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


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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 thecentroid.

2 2( ) =x x yA

I y y dA I Ad


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Moment of Inertia of an I-beam


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Moment of Area