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

dA

ydAy

4

2rA

dA=rdqdr

3

R 1

3

R ydA

cosθ-3

ρ dsin

3

ρddsinρρdρdθρsinθydA

33

π/2

0

R

0

3π/2

0

R

0

3R

0

π/2

0

2

y

qqqrq

dA

x

dA

y=rsinq

x=rcosq

R

r

q

dq

dr

y

3π

4Ryx

3π

4R

4

πR

3

R y

23

dA

ydAx

G

x

y

0y

3

4rxx

G

x

y

3

hy

3

bx

x

y

h

b

h

b

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.

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.

kyy

x

A

O

y

xO

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

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

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

1) RECTANGLE

b

Gx

yb/2

h/2

h

b

y

x

y

y

x

dy

dA=bdy

b

h

dyb

33

3

0

3

0

222 bhbydyybbdyydAyI

hh

x

12232

32322 bhh

bhbh

AdIIh

dAdII xxxx

--

1) RECTANGLE

dx

Gx

yb/2

h/2

h

b

y

x

y

x

x

dA=hdx

h

h

33

3

0

3

0

222 hbhbdxxhhdxxdAxI

hb

y

12

hb

2

bbh

3

hbAeII

2

beAeII

3232

yy

2

yy

--

dx

2. TRIANGLE

Gx

yb/3

h/3

h

b

y

x

h

b

y

x

dyh-y

y

n

y

h

b

yh

n

-From similarity of the triangles,

12

bh

h

b

4

y

3

bydyyh

h

byndyydyI

3h

0

4h

0

3h

0

222

x -- A

36

bh

3

h

2

bh

12

bhAdII

3232

xx

--

yhh

bn -

2. TRIANGLE

Gx

yb/3

h/3

h

b

y

x

h

b

y

xdx

mx

12

hbdAxI

32

y

In a similar manner,

3. SOLID CIRCLE

y

Gx

y

R

y

x

Gx

x

z

y

z

rdr

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/316

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

qq

qq

qq

2cos2sin2

2sin2cos22

2sin2cos22

xyyx

yx

xyyxyx

y

xyyxyx

x

III

I

IIIII

I

IIIII

I

-

-

-

--

qq

qq

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.