sectional properties1

7/24/2019 Sectional Properties1

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Sectional PropertiesSectional Properties


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Sectional Properties-Moment of AreaSectional Properties-Moment of Area

A coplanar surface ofA coplanar surface ofarea A and a referencearea A and a referencexy in the plane of thexy in the plane of the

surface are shown insurface are shown inthe figure . Firstthe figure . Firstmoment of Area Amoment of Area Aabout x axis is definedabout x axis is defined

as:as: MMxx= = AA y dAy dA


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

MMyy= = AA x dAx dA


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!e can sometimes concentrate the entire!e can sometimes concentrate the entirearea A at a point xarea A at a point x cc" y" ycccalled the centroid. #ocalled the centroid. #o

compute these co-ordinates" we e$uate thecompute these co-ordinates" we e$uate themoments of the distributed area with that ofmoments of the distributed area with that ofthe concentrated area.the concentrated area.

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


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yycc= %&A= %&A AA y dAy dA

SimilarlySimilarly

xxcc= %&A= %&A AA x dAx dA


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All axes passing through theAll axes passing through thecentroid are called centroidalcentroid are called centroidal

axes. #he first moments of anaxes. #he first moments of anarea about the centroidal axesarea about the centroidal axesare 'ero.are 'ero.

(onsider a plane area with an(onsider a plane area with anaxes of symmetry )y axis inaxes of symmetry )y axis infigure*. +n e,aluating the integralfigure*. +n e,aluating the integral::

xxcc= %&A= %&A AA x dAx dA

we find that there are a pair ofwe find that there are a pair ofarea elements which are mirrorarea elements which are mirrorimages of each other. #hus ximages of each other. #hus xccisis

'ero.'ero.


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+n many problems" the area of interest can be assumed to be formed by+n many problems" the area of interest can be assumed to be formed bythe addition or subtraction of simple familiar areas whose centroids arethe addition or subtraction of simple familiar areas whose centroids are

nown. !e call areas made up of such simple areas as composite areas.nown. !e call areas made up of such simple areas as composite areas.For such problems" we can say thatFor such problems" we can say that


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Moment of +nertiaMoment of +nertia

Second moment ofSecond moment ofArea A about x axisArea A about x axis)called as moment of)called as moment of

inertia +inertia +xxxx* is defined as:* is defined as: ++xxxx= = AA y y.

.dAdA


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Moment of +nertiaMoment of +nertia

Similarly the second moment of Area ASimilarly the second moment of Area Aabout y axis )called as moment of inertia +about y axis )called as moment of inertia + yyyy**

is defined as:is defined as: ++yyyy= = AA x x.

.dAdA

#he product moment of inertia +#he product moment of inertia + xyxyis definedis defined

as:as: ++xyxy= = AA xy dAxy dA


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+n analogy with the centroid" the entire area may be assumed+n analogy with the centroid" the entire area may be assumedto be concentrated at a single point )to be concentrated at a single point )xx""yy* to gi,e the same* to gi,e the same

moment of inertia of area for a gi,en reference. #hus"moment of inertia of area for a gi,en reference. #hus"


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Moment of +nertia-Parallel AxisMoment of +nertia-Parallel Axis#heorem#heorem

(onsider an aribtrary(onsider an aribtraryx-y system of axes.x-y system of axes.

#he x axis is parallel#he x axis is parallelto an axisto an axis XXgoinggoing

through the centroidthrough the centroidof the Area.of the Area.

#he centroid co-#he centroid co-ordinates w.r.t. x-yordinates w.r.t. x-y

system of axes aresystem of axes are)x)xcc"y"ycc*. +n the next*. +n the nextpage figure they arepage figure they arecalled )c"d*called )c"d*


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Moment of +nertia-Parallel Axis #heoremMoment of +nertia-Parallel Axis #heorem


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Rotated AxisRotated Axis

!e will now obtain second moments and product of area!e will now obtain second moments and product of arearelati,e to a rotated reference. From the figure gi,en below"relati,e to a rotated reference. From the figure gi,en below"

we can see that:we can see that:

x/ = 0 cos )1* 2 3 sin )1*x/ = 0 cos )1* 2 3 sin )1* y/ = -0 sin)1* 2 3 cos)1*y/ = -0 sin)1* 2 3 cos)1* !here 1 is the angle between x and x4 axes!here 1 is the angle between x and x4 axes


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Moment of +nertia-5otated AxisMoment of +nertia-5otated Axis

#hus : +#hus : +x/x/x/x/= = AA y4 y4..dAdA

= = AA)-0 sin)1* 2 3 cos)1**)-0 sin)1* 2 3 cos)1** ..dAdA

6r"6r" ++x/x/x/x/==sinsin.

.))11** AA x x..dAdA

77 sin ) sin )11* cos )* cos )11** AA xy dA xy dA 2 cos2 cos..)1* )1* AA y y.

.dAdA

#herefore"#herefore" ++x/x/x/x/== sinsin.

.))11** ++yyyy7 sin )1* cos )1* +7 sin )1* cos )1* +xyxy2 co2 coss..))11** ++xxxx

Similar results can be obtained for +Similar results can be obtained for + yyyyand +and +xyxy..


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Moment of +nertia-5otated AxisMoment of +nertia-5otated AxisSummari'ed 5esultsSummari'ed 5esults


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!e can compare and see that the transformation relations for!e can compare and see that the transformation relations formoment of inertia and stress transformation relations aremoment of inertia and stress transformation relations are

similar. #he stress transformation relations are gi,en below:similar. #he stress transformation relations are gi,en below:


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Polar moment of inertiaPolar moment of inertia

#he polar moment of#he polar moment ofinertia )denoted byinertia )denoted bysymbol +symbol +ppor 8* isor 8* is

defined as :defined as :++PP= = AA r r.

.dAdA= = AA y y.

.dAdA2 2 AA x x..dAdA

= += +xxxx22 ++yyyy

sectional properties1

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