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Sectional Properties Sectional Properties

Sectional PropertiesSectional Properties--Moment of AreaMoment of Area

A coplanar surface of A coplanar surface of area A and a reference area A and a reference xy in the plane of the xy in the plane of the surface are shown in surface are shown in the figure . First the figure . First moment of Area A moment of Area A about x axis is defined about x axis is defined as:as:

MMxx = = A A y dAy dA

Sectional PropertiesSectional Properties--Moment of AreaMoment of Area

Similarly the First moment of Area A about y Similarly the First moment of Area A about y axis is defined as:axis is defined as:

MMyy = = A A x dAx dA

Sectional PropertiesSectional Properties--Moment of AreaMoment of Area

We can sometimes concentrate the entire We can sometimes concentrate the entire area A at a point xarea A at a point xcc, y, ycc called the centroid. called the centroid. To compute these coTo compute these co--ordinates, we equate ordinates, we equate the moments of the distributed area with that the moments of the distributed area with that of the concentrated area.of the concentrated area.

MMxx = = A A y dA = Ayy dA = Aycc MMyy = = A A x dA= Axx dA= Axcc

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Sectional PropertiesSectional Properties--Moment of AreaMoment of Area

yycc= 1/A= 1/A A A y dA y dA

SimilarlySimilarly

xxcc= 1/A= 1/A A A x dAx dA

All axes passing through the All axes passing through the centroid are called centroidal centroid are called centroidal axes. The first moments of an axes. The first moments of an area about the centroidal axes area about the centroidal axes are zero.are zero.

Consider a plane area with an Consider a plane area with an axes of symmetry (y axis in axes of symmetry (y axis in figure). In evaluating the figure). In evaluating the integral :integral :

xxcc= 1/A= 1/A A A x dAx dAwe find that there are a pair of we find that there are a pair of area elements which are mirror area elements which are mirror images of each other. Thus ximages of each other. Thus xcc is is zero. zero.

In many problems, the area of interest can be assumed to be formed by In many problems, the area of interest can be assumed to be formed by the addition or subtraction of simple familiar areas whose centroids are the addition or subtraction of simple familiar areas whose centroids are known. We call areas made up of such simple areas as composite areas. known. We call areas made up of such simple areas as composite areas. For such problems, we can say thatFor such problems, we can say that

Moment of InertiaMoment of Inertia Second moment of Second moment of

Area A about x axis Area A about x axis (called as moment of (called as moment of inertia Iinertia Ixxxx) is defined ) is defined as:as:

IIxxxx = = AA yy22 dAdA

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Moment of InertiaMoment of Inertia Similarly the second moment of Area A Similarly the second moment of Area A

about y axis (called as moment of inertia Iabout y axis (called as moment of inertia Iyyyy) ) is defined as:is defined as:

IIyyyy = = AA xx22 dAdA The product moment of inertia IThe product moment of inertia Ixyxy is defined is defined

as:as:

IIxyxy = = AA xy dAxy dA

In analogy with the centroid, the entire area may be assumed In analogy with the centroid, the entire area may be assumed to be concentrated at a single point (kto be concentrated at a single point (kxx,k,kyy) to give the same ) to give the same moment of inertia of area for a given reference. Thus,moment of inertia of area for a given reference. Thus,

Moment of InertiaMoment of Inertia--Parallel Axis Parallel Axis TheoremTheorem

Consider an aribtrary Consider an aribtrary xx--y system of axes. y system of axes.

The x axis is parallel The x axis is parallel to an axis to an axis XXXXXXXX going going through the centroid through the centroid of the Area. of the Area.

The centroid coThe centroid co--ordinates w.r.t. xordinates w.r.t. x--y y system of axes are system of axes are (x(xcc,y,ycc). In the next ). In the next page figure they are page figure they are called (c,d)called (c,d)

Moment of InertiaMoment of Inertia--Parallel Axis TheoremParallel Axis Theorem

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Rotated AxisRotated AxisWe will now obtain second moments and product of area We will now obtain second moments and product of area relative to a rotated reference. From the figure given below, relative to a rotated reference. From the figure given below, we can see that:we can see that:

x' = X cos () + Y sin ()x' = X cos () + Y sin () y' = y' = --X sin() + Y cos()X sin() + Y cos() Where is the angle between x and x axesWhere is the angle between x and x axes

Moment of InertiaMoment of Inertia--Rotated AxisRotated Axis Thus : IThus : Ix'x'x'x' = = AA yy22 dAdA

= = AA((--X sin() + Y cos())X sin() + Y cos()) 22 dAdA Or, Or, IIx'x'x'x' ==sinsin22 (() ) AA xx22 dAdA 2 sin (2 sin () cos () cos () ) AA xy dAxy dA + cos+ cos22 () () AA yy22 dAdA

Therefore,Therefore, IIx'x'x'x' = = sinsin22 (() ) IIyy yy 2 sin () cos () I2 sin () cos () Ixyxy + co+ coss22 (() ) IIxxxx Similar results can be obtained for ISimilar results can be obtained for Iyyyy and Iand Ixyxy. .

Moment of InertiaMoment of Inertia--Rotated AxisRotated AxisSummarized ResultsSummarized Results

We can compare and see that the transformation relations for We can compare and see that the transformation relations for moment of inertia and stress transformation relations are moment of inertia and stress transformation relations are

similar. The stress transformation relations are given below:similar. The stress transformation relations are given below:

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Polar moment of inertiaPolar moment of inertia The polar moment of The polar moment of

inertia (denoted by inertia (denoted by symbol Isymbol Ipp or J) is or J) is defined as :defined as :

IIPP = = AA rr22 dAdA= = AA yy22 dAdA + + AA xx22 dAdA= I= Ixxxx+ + IIyyyy