centroid (aĞirlik merkezİ ) · 2014-06-02 · moment of inertia is not a physical quantity such...
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CENTROID (AĞIRLIK MERKEZİ )
A centroid is a geometrical concept arising from parallel forces. Thus,
only parallel forces possess a centroid. Centroid is thought of as the
point where the whole weight of a physical body or system of particles is
lumped. If proper geometrical bodies possess an axis of symmetry, the
centroid will lie on this axis. If the body possesses two or three
symmetry axes, then the centroid will be located at the intersection of
these axes.
If one, two or three dimensional bodies are defined as analytical
functions, the locations of their centroids can be calculated using
integrals.
A composite body is one which is comprised of the combination of several simple
bodies. In such bodies, the centroid is calculated as follows:
Line-a thin rod
(Çizgi)
Area-a flat plate with constant
thickness (Alan)
Volume-a sphere or a cone
(Hacim)
Composite Composite Composite
dl
xdlx
dl
ydly
dl
zdlz
i
ii
i
ii
i
ii
l
lzz
l
lyy
l
lxx
dA
zdAz
dA
ydAy
dA
xdAx
i
ii
i
ii
i
ii
A
Azz
A
Ayy
A
Axx
dV
zdVz
dV
ydAy
dV
xdVx
i
ii
i
ii
i
ii
V
Vzz
V
Vyy
V
Vxx
Thin Rod in the Shape of a Quarter Circle (Çeyrek Çember Şeklinde İnce Çubuk)
r=radius
dA
ydAy
4
rA
2
dA=rdqdr
3π
4Ryx
3π
4R
πR
4
3
R
3
R 1-0-
3
R cosθ-
3
ρ
dsin3
ρddsinρρsinθρdρdθydA
2
3
33π/2
0
R
0
3
π/2
0
R
0
3R
0
π/2
0
2
qqqrq
x
dA
y=rsin
q
x=rcosq
R
r
q
dq
dr
y
G
x
y
0y
3
r4x
x
G
x
y
3
hy
3
bx
x
y
h
b
h
b
w=b-x
y
x
b
x
w=b-x
b/2
y
dy
dx
dy
y
dx
yupper
ylower
xright
xleft
x
y
623
2 2
0
2
0
23
0
212121
2121
axxadxxxadAA
dxxxadxyyldxdA
dA
xdAx
aa
/a
///
//lowerupper
--
--
5
2
6
15
1535
2
2
3
3
0
3
0
25
0
212121 a
a
a
xax
xadxxxaxxdA
aa
/a
/// --
dx
l
y2=ax y=a1/2x1/2
yupper
ylower
xright
xleft
x
y
2
6
12
12434
1
3
2
3
333
0
4
0
3
0
2
a
a
a
y
aaay
a
ydy
a
yyydyxxyywdyydA
aaa
leftright
--
--
dy
w
dA
ydAy
6
2aA
y2=ax x=y2/a
It is often necessary to calculate the moments of uniformly distributed
loads about an axis lying within the plane they are applied to or
perpendicular to this plane. Generally, the magnitudes of these forces per
unit area (pressure or stress) are proportional to distance of the line action
of the force from the moment axis. The elemental force acting on an
element of area, then is proportional to distance times differential area,
and the elemental moment is proportional to distance squared times
differential area.
Thus, the total moment: dM=M=d2dA.
This integral is named as “Area Moment of Inertia” or
“Second Moment of Area”.
Elemantary moment is proportional to distance2 × differential area:
dM=d2dA
Moment of inertia is not a physical quantity such as velocity, acceleration or
force, but it enables ease of calculation; it is a function of the geometry of the
area. Since in Dynamics there is no such concept as the inertia of an area, the
moment of inertia has no physical meaning.
But in mechanics, moment of inertia is used in the calculation of bending of a
bar, torsion of a shaft and determination of the stresses in any cross section of a
machine element or an engineering structure.
Ix=y2dA Inertia moment of area A with respect to x-axis
Iy=x2dA Inertia moment of area A with respect to y-axis
Rectangular Moments of Inertia
Polar Moments of Inertia
Io=Iz=r2dA r2=x2+y2
Io=Iz= Ix+ Iy
Product of Inertia
(Çarpım Alan Atalet Momenti)
Ixy=xydA
In certain problems involving unsymmetrical cross sections and in calculation of
moments of inertia about rotated axes, an expression dIxy=xydA occurs, which
has the integrated form
Properties of moments of inertia :
1. Area moments of inertia Io, Ix , Iy are always positive .
3. The unit for all area moments of inertia is the 4. power of that
taken for length (L4).
2. Ixy may be (-), (+) or zero whenever either of the
reference axes is an axis of symmetry, such as the x axis
in the figure.
Properties of moments of inertia :
4. The smallest value of an area moment of inertia that an area can
have is realized with respect to an axis that passes from the centroid
of this area. The area moment of inertia of an area increases as the
area goes further from this axis.
The area moment of inertia will get smaller when the distribution of
an area gets closer to the axis as possible.
Jirasyon (Atalet – Eylemsizlik) Yarıçapı
Consider an area A, which has rectangular moment of inertia Ix. We now
visualize this area as concentrated into a long narrow strip of area A
a distance kx from the x axis. By definition, the moment of inertia of the
strip about the x axis will be the same as that of the original area if
Ix=kx2A
The distance kx is called the “radius of gyration” of the area about the x
axis.
A
Ik x
x kx
y
x
A
O
y
O
A
z
x
Radii of gyration about the y-and z axes are obtained in the same manner.
ky y
x
A
O
y
x O
kz
A
AkI A kI2
zz
2
yy
A
Ik
y
y A
Ikk o
oz
2y
2x
2zzyx kkk III Also since,
G
x
y
O
d
e
A
r
x
y
The moment of inertia of an area about a noncentroidal axis may be easily
expressed in terms of the moment of inertia about a parallel centroidal axis.
AdeII
ArII
AeII
AdII
xyxy
2zz
2yy
2xx
2
2
AeII
AdII
yy
xx
The sum of these two equations gives
2ArII zz zyxzyx IIIIII ,and since
G
x
y
O
d
e
A
r
AdeII xyxy
For product of inertia
222rkk zz
where is the radius of gyration about a centroidal axis parallel to
the axis about which k applies and r is the perpendicular distance
between the two axes. For product of inertia:
dekk xyxy 22
k
The Parallel-axis theorems also hold for radii of gyration as:
dekk 22
Two points that should be noted in particular about the transfer of
axes are:
The two transfer axes must be parallel to each other
One of the axes must pass through the centroid of the area
1) RECTANGLE
b
G
x
y b/2
h/2
h
b
y
x
y
y
x
dy
dA=bdy
b
h
dy b
3
bh
3
bydyybbdyydAyI
3h
0
3h
0
222x
12
bh
2
hbh
3
bhAdII
2
hdAdII
3232
xx
2
xx
--
1) RECTANGLE
dx
G
x
y b/2
h/2
h
b
y
x
y
x
x
dA=hdx
h
h
3
hb
3
hbdxxhhdxxdAxI
3h
0
3b
0
222y
12
hb
2
bbh
3
hbAeII
2
beAeII
3232
yy
2
yy
--
dx
2. TRIANGLE
G x
y b/3
h/3
h
b
y
x
h
b
y
x
dy h-y
y
n
y
h
b
yh
n
-From similarity of the triangles,
12
bh
h
b
4
y
3
bydyyh
h
byndyyAdyI
3h
0
4h
0
3h
0
222
x --
36
bh
3
h
2
bh
12
bhAdII
3232
xx
--
yhh
bn -
2. TRIANGLE
G x
y b/3
h/3
h
b
y
x
h
b
y
x dx
m x
12
hbdAxI
32
y
In a similar manner,
3. SOLID CIRCLE
y
G x
y
R
y
x
G x
x
z
y
z
r dr
2
πR
4
R2
4
r2πI
drr2π rdr2rI rdr 2dA dArI
44R
0
4
z
R
0
32
z
2
z
yxzo IIII
Due to symmetry; 4
πRII
4
yx
4. SEMI CIRCLE
G x
y
O x
4R/3
8
R
2
4
R
I4
4
x
-
-
-
9
8
8R
3
R4
2
R
8
RAdII 4
2242
xx
8
R
2
4
R
I4
4
y
5. QUARTER CIRCLE
y
x
x
y
G
4R/3
4R/3 16
R
4
4
R
II4
4
yx
-
--
9
4
163
4
416
4224
2 RRRR
AdII xx
-
--
9
4
163
4
416
4224
2 RRRR
AeII yy
In Mechanics it is often necessary to calculate the moments of
inertia about rotated axes.
The product of inertia is useful when we need to calculate the
moment of inertia of an area about inclined axes. This
consideration leads directly to the important problem of
determining the axes about which the moment of inertia is a
maximum and a minimum.
Given dAxyI,dAxI,dAyI xyyx22
2cos2sin2
2sin2cos22
2sin2cos22
xyyx
yx
xyyxyx
y
xyyxyx
x
III
I
IIIII
I
IIIII
I
-
-
-
--
sincos
sincos
xyy
yxx
-
Note:
We wish to determine moments and product of inertia with respect to new
axes x and y.
22
minmax 4
2
1
2xyyx
yxIII
III -
yx
xym
II
I
--
22tan q
Imax and Imin are the principal moments of inertia.
The product of inertia is zero for the principle axes of inertia.
The equation for qm defines two angles 90o apart which correspond to the principal
axes of the area about O.
Sample Problem: Determine the orientation of the
principal axes of the inertia through the centroid the
angle section and determine the corresponding maximum
and minimum moments of inertia.
50
mm
40 mm
10 mm
10 mm
7.5 mm G
2.5 mm
The product of inertia about the x-y axes for part I is;
40 mm
10 mm
10 mm
7.5 mm
G
2.5 mm
I
II
G1
G2
44
1
1
1075.3
)1040(5.2102055.70
111
mmI
AyxII
xy
Add
xyxy
yx
-
---
The product of inertia about the x-y axes for part II is;
44
2
2
105.7
)1040(5.255.7200
222
mmI
AyxII
xy
Add
xyxy
yx
-
--
dy1
dx1
dy2
dx2 50
mm
x
y
x
y
For the complete section;
Ixy=-7.5.104 mm4
The moments of inertia about the x and y axes for part I is;
40 mm
10 mm
10 mm
7.5 mm
G
2.5 mm
I
II
G1
G2
44232
1
442321
1058.75.7400401012
1
1058.65.12400104012
1
mmAeII
mmAdII
yy
xx
dy1
dx1
dx2
dy2 50
mm
x
y
The moments of inertia about the x and y axes for part II is;
44232
2
442322
1058.25.7400104012
1
1058.115.12400401012
1
mmAeII
mmAdII
yy
xx
For the complete section;
Ixy=-7.5.104 mm4
Ix=18.17.104 mm4 Iy=10.17.104 mm4
om
yx
xy
mII
I31
17.1817.10
)5.7(222tan
-
--
44min
44max
22
minmax
1067.5
107.2242
1
2
mmI
mmIIIIII
I xyyx
yx
-