11459_PROPERTIES of Plane Surfaces

8/7/2019 11459_PROPERTIES of Plane Surfaces

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PROPERTIES OF PLANE

SURFACESBY Dr.A.K.Srivastava


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First Moment of Area

yi

xi

x

y

o

Ai

!(!i

iix ydAAyM

i

iiy xdAAxM


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Centroid

A

M

A

xdAX

y

C !!

A

M

A

ydA

Yx

C !!


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CENTROID OF A SURFACE MADE BY JOINING MANY

DIFFERENT SURFACES

!!

i

i

i

iCi

total

i

iCi

CA

AX

A

AX

X

!!

ii

i

iCi

total

i

iCi

C

A

AY

A

AY

Y


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APPLICATIONS TO MECHANICS

f(x)R

X1 X2

X

dF=f(x)dx

!2

1

)(

x

x

dxxfR

Rdxxxf

x

x

!2

1

)(

XdxxxfR

x

x

!2

1

)(1


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Uniform Loading

X1 X2

x

y

w

f(x)

(X1+X2)/2

R=w(X2-X1)

2

)(

2

)( 21121

XXXXXX

C

!

!


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Triangular Loading

X1 X2

(X1+X2)/2w

R

2

)( 12 XXwR

!

3

)2(

3

)(2 21121

XXXXXXC

!

!

f(x)

3/)( baXc !


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X1 X2

w2w1

f(x)

x

)(

)2()2(

3

1

21

212211

ww

wwXwwXX

C

!

Trapezoidal Loading


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Plane Surface Submerged In Water

x

y

h1

h2

dy dy/cos

l

The rectangular plate has

length l and width w and

is submerged in water at

an angle from the

vertical. The upper end of

the plate is at the depth of

h1 and the lower one at

the depth of h2 .


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At depth y the pressure acting on the plate=gy

Now take a thin strip of width dy/cos at depth y parallel to the

plates width, its area is dA= wdy/cos and the force on itwould be dF= gywdy/cos .

The total force on the plate would be

The average pressure

! U

Vcos

gywdy

F

platetheofArea

gywdy

Paverage ! U

Vcos


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Introduce another length variable Y along the plate.

=Y distance of the centroid of the plate

Ucos

yY

!

UV

cos

platetheofArea

gYwdYPaverage

!

cosCaverage gY!

CY

x

y Y


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centroi

daver

age

gyV!

The average pressure on the plate is g(the depth of the centroid of the plate).

This is true for a planar surface of any shape

)()(

32

21

2

221

2

1

hhhhhh

For a rectangular plate the loading curve is a trapezoid. The

total force acts at a depth of


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SECOND MOMENT OF AREA

!(!i

iXX dAyAyI22

!(! i AiYY

dAxAxI22

!(!

i AiiXY

xydAAyxI

yi

xi

o


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Transfer Theorem

AyII XXXX2

0'' !AxII YYYY

2

0!

AyxII YXXY 00'' !

O

2a

2b

X

Y

x

y

o

y0

x0,y0)

x0

!! dAyydAyIXX

2

0

'2 )(

! dAyydAydAy '2 0202'


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Transformation of Moments and Products of Area

x

y xy

O

UU sincos' yxx !

UU cossin' yxy !


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!! UUU 2sincossin' 222'' XYXXYYXX IIIdAyI

)2cos1(2

1sin

2 UU ! )2cos1(2

1cos

2 UU !

UU 2sin2cos22'' XY

YYXXYYXXXX I

IIIII

!

!! UUU 2sinsincos' 222'' XYXXYYYY IIIdAxI

UU 2sin2cos22

'' XYYYXXYYXX

YY IIIII

I

!


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UU 2cos2sin2'' XYYYXX

YX III

I

!

! dAyxI YX ''''


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Principle Axes

The principle set of axes at a point are those for which theproduct of inertia vanishes i.e. IXY=0 about the principle set of

axes.

If the principle set of axes (1,2) are obtained by rotating the x-y

axes by angle ,the we want I12 to vanish.

02cos2sin2

12 !

! EE XYYYXX III

I

XXYY

XY

III

! 22tan E


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The principle set of axes have one more property-the moment

of area is maximum about one of the principle axis (say x-axis)

and minimum about the other. We know that

Let us find for which IXX is a maximum or a minimum.

UU 2sin2cos22

'' XYYYXXYYXX

XX IIIII

I

!

0'' !

x

x

U

XXI

XXYY

XY

II

I

!

22tan U


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This is the same angle that makes IXY vanish. This means

2=2 or 2+

= or (+/2)

When makes the function IXX a maximum, the angle (+/2)

makes IYY a minimum. Thus , the principle set of axes are also

those about which the second moment of area is maximum

about one axis and minimum about the other.


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Derivation


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11459_PROPERTIES of Plane Surfaces