11459_PROPERTIES of Plane Surfaces
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Transcript of 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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