Mechsnics Of Material by Hibbler

1. Centroids & Moment of
Inertia
MMT -I
1

2. Centroid or Center of Area
Centroid: A point at which whole area of body or a plane figure is acting is called
centroid. Also known as geometric center
• It is only applicable to plane figures having certain area but no volume, for
example; rectangle, square, circle, semi-circle, triangle, etc.
Composite Figure made up of two or more plane figure such as triangles,
rectangles, circles, semi-circles, etc
Center of Gravity (COG) point at which the whole mass of body is concentrated.
COG and centroid are different quantities but both become similar when bodies have
only area but not weight.
2

3. Plane Figures vs Composite Figures
3

4. Centroid vs Center of Gravity (COG)
Whole weight is acting
at this point
Other Examples = Book or pen balanced on finger
 Suspend the plane fig from corners
 From suspension point, draw a vertical line that will be
directed towards the earth
 Repeat the step at other corners also.
 Intersection point of lines will be the centroid of fig
4

5. Centroid vs Center of Gravity (COG)
Steel Plate
Steel +Aluminum
Plate
COG
Centroid Centroid
COG
Entire area of plane figure is
concentrated
Entire weight is assumed to be
concentrated
5

6. 6

7. Centroid Location of a composite figure
• Place x axis at lowest point and y axis at the left edge
of figure (first quadrant, where X & Y will be +ive).
• Split the composite figure into plane figure & find
centroid of each plane figure
• Now following formulas can be used to find the X
and Y coordinates of centroid of composite figure.
• 𝒙𝒄 =
𝑨𝟏𝒙𝟏+𝑨𝟐𝒙𝟐 …….𝑨𝒏𝒙𝒏
𝑨𝟏+𝑨𝟐 …….𝑨𝒏
,
• 𝒚𝒄 =
𝑨𝟏𝒚𝟏+𝑨𝟐𝒚𝟐 …….𝑨𝒏𝒚𝒏
𝑨𝟏+𝑨𝟐 …….𝑨𝒏
• x & y in above formula mean the dimensions along x & y axis respectively.
• Locate the centroid on figure
7
𝑪 𝒙𝒄, 𝒚𝒄

8. Figure having one axis of symmetry:
• Any figure having one axis of symmetry, its centroid
will lie at that axis.
• Centroid of figure will lie on x-axis because it is
symmetric about x axis. Therefore only one coordinate
must be calculated to locate the centroid.
Figure having two axis of symmetry:
Any figure having two axis of symmetry, then its
centroid will lie at the intersection of axis of symmetry.
8

9. Figure having no axis of symmetry:
• Any figure having no axis of symmetry, its
centroid will be located by inspection.
• Such figure is only symmetric about a point,
called the center of symmetry, such that every
line drawn through that point contacts the area
in a symmetrical manner.
9

10. Q. Determine the centroid of C
section beam.
X3
X2
X1
y3
y2
y1
Area Value
𝒄𝒎𝟐
X-
coord
inate
Value
(cm)
Y-
coord
inate
Value
(cm)
𝒚(cm)
𝒚𝒏 − 𝒚𝒄
𝒙 (cm)
𝒙𝒏 − 𝒙𝒄
𝑨𝟏 7.5 𝒙𝟏 2.5 𝒚𝟏 0.75 𝑦1 = ? 𝑥1 = ?
𝑨𝟐 10.5 𝒙𝟐 0.75 𝒚𝟐 5 𝑦2 = ? 𝑥2 = ?
𝑨 7.5 𝒙𝟑 2.5 𝒚𝟑 9.25 𝑦3 = ? 𝑥3 = ?
10

11. 𝒄𝟏 𝟐. 𝟓, 𝟎. 𝟕𝟓
𝒄𝟐 𝟎. 𝟕𝟓, 𝟓
𝒄𝟑 𝟐. 𝟓, 𝟗. 𝟐𝟓
𝑪 𝟏. 𝟕𝟖, 𝟓
Area Value
𝒄𝒎𝟐
X-
coord
inate
Value
(cm)
Y-
coord
inate
Value
(cm)
𝒚(cm)
𝒚𝒏 − 𝒚𝒄
𝒙 (cm)
𝒙𝒏 − 𝒙𝒄
𝑨𝟏 7.5 𝒙𝟏 2.5 𝒚𝟏 0.75 𝑦1 = ? 𝑥1 = ?
𝑨𝟐 10.5 𝒙𝟐 0.75 𝒚𝟐 5 𝑦2 = ? 𝑥2 = ?
𝑨𝟑 7.5 𝒙𝟑 2.5 𝒚𝟑 9.25 𝑦3 = ? 𝑥3 = ?
11

12. Q. Determine the centroid of I
section beam.
I section is symmetric about y axis, so centroid will
lie along y axis.
x
y
12
Area Value
𝒄𝒎𝟐
X-
coord
inate
Value
(cm)
Y-
coord
inate
Value
(cm)
𝒚(cm)
𝒚𝒏 − 𝒚𝒄
𝒙 (cm)
𝒙𝒏 − 𝒙𝒄
𝑨𝟏 150 𝒙𝟏 15 𝒚𝟏 2.5 𝑦1 = ? 𝑥1 = ?
𝑨𝟐 75 𝒙𝟐 15 𝒚𝟐 12.5 𝑦2 = ? 𝑥2 = ?
𝑨𝟑 100 𝒙𝟑 15 𝒚𝟑 22.5 𝑦3 = ? 𝑥3 = ?

13. x
y
13
𝒄𝟏 𝟏𝟓, 𝟐. 𝟓
𝒄𝟐 𝟏𝟓, 𝟏𝟐. 𝟓
𝒄𝟑 𝟏𝟓, 𝟐𝟐. 𝟓
𝑪 𝟏𝟓, 𝟏𝟎. 𝟗𝟔
Area Value
𝒄𝒎𝟐
X-
coord
inate
Value
(cm)
Y-
coord
inate
Value
(cm)
𝒚(cm)
𝒚𝒏 − 𝒚𝒄
𝒙 (cm)
𝒙𝒏 − 𝒙𝒄
𝑨𝟏 150 𝒙𝟏 15 𝒚𝟏 2.5 𝑦1 = ? 𝑥1 = ?
𝑨𝟐 75 𝒙𝟐 15 𝒚𝟐 12.5 𝑦2 = ? 𝑥2 = ?
𝑨𝟑 100 𝒙𝟑 15 𝒚𝟑 22.5 𝑦3 = ? 𝑥3 = ?

14. Q. Determine the Centroid of Z
section beam.
14
Area Value
𝒄𝒎𝟐
X-
coord
inate
Value
(cm)
Y-
coord
inate
Value
(cm)
𝒚(cm)
𝒚𝒏 − 𝒚𝒄
𝒙 (cm)
𝒙𝒏 − 𝒙𝒄
𝑨𝟏 100 𝒙𝟏 17.5 𝒚𝟏 2.5 𝑦1 = ? 𝑥1 = ?
𝑨𝟐 37.5 𝒙𝟐 8.75 𝒚𝟐 12.5 𝑦2 = ? 𝑥2 = ?
𝑨𝟑 25 𝒙𝟑 5 𝒚𝟑 21.25 𝑦3 = ? 𝑥3 = ?

15. 15
Area Value
𝒄𝒎𝟐
X-
coord
inate
Value
(cm)
Y-
coord
inate
Value
(cm)
𝒚(cm)
𝒚𝒏 − 𝒚𝒄
𝒙 (cm)
𝒙𝒏 − 𝒙𝒄
𝑨𝟏 100 𝒙𝟏 17.5 𝒚𝟏 2.5 𝑦1 = ? 𝑥1 = ?
𝑨𝟐 37.5 𝒙𝟐 8.75 𝒚𝟐 12.5 𝑦2 = ? 𝑥2 = ?
𝑨𝟑 25 𝒙𝟑 5 𝒚𝟑 21.25 𝑦3 = ? 𝑥3 = ?
𝒄𝟏 𝟏𝟕. 𝟓, 𝟐. 𝟓
𝒄𝟐 𝟖. 𝟕𝟓, 𝟏𝟐. 𝟓
𝒄𝟑 𝟓, 𝟐𝟏. 𝟐𝟓
𝑪 𝟏𝟑. 𝟓𝟓, 𝟕. 𝟔𝟗

16. Moment of force or 1st Moment of force: Moment of force about any point = product of force and
perpendicular distance between the point and line of action of force.
2nd Moment of force: 1st moment of force again multiplied by perpendicular distance gives quantity
called 2nd moment of force. It can never be zero due to perpendicular distance between two axis.
𝟐𝒏𝒅 𝐌𝐨𝐦𝐞𝐧𝐭 𝐨𝐟 𝐟𝐨𝐫𝐜𝐞 = 𝑭𝒅𝟐
1st Moment of area or 1st MOI: Area of figure multiplied by perpendicular distance from reference axis
is called 1st moment of area. It is the measure of area distribution of a shape in relation to an axis.
Mathematically, 𝟏𝒔𝒕
Moment of area = area ∗ perpendicular distance
Units = (length)3
2nd Moment of area/mass or MOI: Area of figure or mass of body multiplied by square of
perpendicular distance from reference axis is called 2nd moment of area/mass.
Moment of Inertia (MOI): 2nd moment of area/mass is broadly termed as MOI. 16

17. Moment of Inertia (MOI): MOI is always a +ive quantity. It is denoted by I
Units = m4 or mm4 or (length)4
Mass moment of inertia: When mass MOI is used in combination with rotation of rigid
bodies, it can be considered as “Measure of Body’s Resistance To Rotation”.
Area moment of inertia: When area MOI is used in combination with deflection or
deformation of members in bending, it can be considered as “Measure of Body’s
Resistance To Bending”.
Polar moment of inertia: When polar MOI is used in combination with member
subjected to torsional loadings, it can be considered as “Measure of Body's Resistance To
Torsion”.
MOI form the basis of dynamics of rigid bodies and strength of materials.
17

18. MOI of rectangle about centroidal X-axis:
𝐼𝑥𝑥 =
𝑏𝑑3
12
MOI of rectangle about centroidal Y-axis:
𝐼𝑦𝑦 =
𝑑𝑏3
12
x
x
y
y
C
d
b
MOI changes when axis is changed.
In above formula,
dimension parallel to centroidal axis will be taken as b (width of beam) and
dimensions perpendicular to centroidal axis will be taken as d (depth of beam).
Once b & d are fixed during horizontal axis then never change it, during finding
MOI about another vertical axis 18

19. MOI of Hollow Rectangular section:
MOI of outer rectangle =
𝑏𝑑3
12
MOI of inner rectangle =
𝑏1𝑑1
3
12
𝑰𝒙𝒙 =
𝒃𝒅𝟑
𝟏𝟐
−
𝒃𝟏𝒅𝟏
𝟑
𝟏𝟐
Similarly, 𝑰𝒚𝒚 =
𝒅𝒃𝟑
𝟏𝟐
−
𝒅𝟏𝒃𝟏
𝟑
𝟏𝟐
When centroid or center of gravity of
both rectangles concide each other
d d1
19

20. MOI of circular section:
Now MOI about centroidal axis which is
perpendicular to circular cross section
𝐼𝑥𝑥 = 𝐼𝑦𝑦 =
𝜋
64
(𝑑)4 =
𝜋
4
(𝑟)4
MOI of hollow circular cross section:
𝐼𝑥𝑥 = 𝐼𝑦𝑦 =
𝜋
64
(𝐷4
− 𝑑4
)
x
x
y
y
C
r
x
x
y
y
C
20

21. 21

22. MOI of Composite Figure
Moment of inertia of composite section is find out by following steps
1. After drawing axis, split the given composite figure into plane figure
(rectangular, circular, triangular).
2. Find the centroid of each plane figure
3. Find the X & Y coordinates of centroid of Composite figure
4. Now use Parallel axis theorem to find MOI about the required axis
(or centroidal axis of Composite figure)..
22

23. Parallel Axis Theorem
MOI of area with respect to any axis in its plane = Centroidal MOI about
centroidal axis of small plane figure + the product of area and square of
distances between these two parallel axis.
Mathematically,
MOI w.r.t to X-axis;
𝑰𝒙𝒙 = 𝑰𝒄𝒙 + 𝑨(𝒚)𝟐
𝒙 &𝒚 = 𝐝𝐢𝐬𝐭𝐚𝐧𝐜𝐞 𝐛𝐞𝐭𝐰𝐞𝐞𝐧 𝐭𝐰𝐨 𝐩𝐚𝐫𝐫𝐚𝐥𝐞𝐥 𝐚𝐱𝐢𝐬
𝐈𝒄𝒙 & 𝐈𝐜𝐲 = 𝐌𝐎𝐈 𝐨𝐟 𝐬𝐦𝐚𝐥𝐥 𝐟𝐢𝐠. 𝐚𝐛𝐨𝐮𝐭 𝐢𝐭𝐬 𝐨𝐰𝐧 𝐜𝐞𝐧𝐭𝐫𝐨𝐢𝐝𝐚𝐥 𝐚𝐱𝐢𝐬 𝐨𝐫 𝐜𝐞𝐧𝐭𝐫𝐨𝐢𝐝𝐚𝐥 𝐌𝐎𝐈
MOI w.r.t Y-axis
𝑰𝒚𝒚 = 𝑰𝒄𝒚 + 𝑨(𝒙)𝟐
23

24. If MOI about centroidal axis (of small plane figure) is known, we can find the
moment of inertia about any required axis (or about the centroid of composite
figure) by using Parallel Axis Theorem.
• From Parallel Axis Theorem, it is clear that MOI increases as reference axis
is moved parallel and farther away from centroid.
• MOI about centroidal axis is least moment of inertia.
• When using Parallel axis theorem, always remember that one of two parallel
axis must be centroidal axis.
24

25. MOI of rectangle about centroidal X-axis:
𝐼𝑐𝑥 =
𝑏𝑑3
12
MOI of rectangle about centroidal Y-axis:
𝐼𝑐𝑦 =
𝑑𝑏3
12
x
x
y
y
C
A B
C
D
d
b
 Icx & Icy centroidal MOI or MOI of small
figure about its own centroidal axis
 𝒙 & 𝒚 = distance between two parallel axis.
𝒙
𝒚
MOI w.r.t to X-axis;
𝑰𝑨𝑩 = 𝑰𝒄𝒙 + 𝑨(𝒚)𝟐
MOI w.r.t Y-axis
𝑰𝑪𝑫 = 𝑰𝒄𝒚 + 𝑨(𝒙)𝟐 25

26. 26

27. Q. Determine the MOI of C section
beam w.r.t its centroidal axis.
X3
X2
X1
y3
y2
y1
27
Area Value
𝒄𝒎𝟐
X-
coord
inate
Value
(cm)
Y-
coord
inate
Value
(cm)
𝒚(cm)
𝒚𝒏 − 𝒚𝒄
𝒙 (cm)
𝒙𝒏 − 𝒙𝒄
𝑨𝟏 7.5 𝒙𝟏 2.5 𝒚𝟏 0.75 𝑦1 = ? 𝑥1 = ?
𝑨𝟐 10.5 𝒙𝟐 0.75 𝒚𝟐 5 𝑦2 = ? 𝑥2 = ?
𝑨𝟑 7.5 𝒙𝟑 2.5 𝒚𝟑 9.25 𝑦3 = ? 𝑥3 = ?

28. 𝒄𝟏 𝟐. 𝟓, 𝟎. 𝟕𝟓
𝒄𝟐 𝟎. 𝟕𝟓, 𝟓
𝒄𝟑 𝟐. 𝟓, 𝟗. 𝟐𝟓
𝑪 𝟏. 𝟕𝟖, 𝟓
28
Area Value
𝒄𝒎𝟐
X-
coord
inate
Value
(cm)
Y-
coord
inate
Value
(cm)
𝒚(cm)
𝒚𝒏 − 𝒚𝒄
𝒙 (cm)
𝒙𝒏 − 𝒙𝒄
𝑨𝟏 7.5 𝒙𝟏 2.5 𝒚𝟏 0.75 𝑦1 = ? 𝑥1 = ?
𝑨𝟐 10.5 𝒙𝟐 0.75 𝒚𝟐 5 𝑦2 = ? 𝑥2 = ?
𝑨𝟑 7.5 𝒙𝟑 2.5 𝒚𝟑 9.25 𝑦3 = ? 𝑥3 = ?

29. 𝑰𝒙𝒙𝟐 = 𝑰𝒄𝒙𝟐
+ 𝑨𝟐(𝒚)𝟐
𝑰𝒙𝒙𝟑 = 𝑰𝒄𝒙𝟑
+ 𝑨𝟑(𝒚)𝟐
𝑰𝒙𝒙𝟏 = 𝑰𝒄𝒙𝟏
+ 𝑨𝟏(𝒚)𝟐
b
d
𝑰𝒄𝒙
𝑰𝒄𝒚
29

30. 𝑰𝒚𝒚𝟏 = 𝑰𝒄𝒚𝟏
+ 𝑨𝟏(𝒙)𝟐
𝑰𝒚𝒚𝟐 = 𝑰𝒄𝒚𝟐
+ 𝑨𝟐(𝒙)𝟐
30
b
d
𝑰𝒄𝒙 𝑰𝒄𝒚
𝑰𝒚𝒚𝟑 = 𝑰𝒄𝒚𝟑
+ 𝑨𝟑(𝒙)𝟐

31. Q. Determine the MOI of I section
beam about its centroidal axis
I section is symmetric about y axis, so centroid will
lie along y axis.
x
y
31
Area Value
𝒄𝒎𝟐
X-
coord
inate
Value
(cm)
Y-
coord
inate
Value
(cm)
𝒚(cm)
𝒚𝒏 − 𝒚𝒄
𝒙 (cm)
𝒙𝒏 − 𝒙𝒄
𝑨𝟏 150 𝒙𝟏 15 𝒚𝟏 2.5 𝑦1 = ? 𝑥1 = ?
𝑨𝟐 75 𝒙𝟐 15 𝒚𝟐 12.5 𝑦2 = ? 𝑥2 = ?
𝑨𝟑 100 𝒙𝟑 15 𝒚𝟑 22.5 𝑦3 = ? 𝑥3 = ?

32. x
y
32
𝒄𝟏 𝟏𝟓, 𝟐. 𝟓
𝒄𝟐 𝟏𝟓, 𝟏𝟐. 𝟓
𝒄𝟑 𝟏𝟓, 𝟐𝟐. 𝟓
𝑪 𝟏𝟓, 𝟏𝟎. 𝟗𝟔
Area Value
𝒄𝒎𝟐
X-
coord
inate
Value
(cm)
Y-
coord
inate
Value
(cm)
𝒚(cm)
𝒚𝒏 − 𝒚𝒄
𝒙 (cm)
𝒙𝒏 − 𝒙𝒄
𝑨𝟏 150 𝒙𝟏 15 𝒚𝟏 2.5 𝑦1 = ? 𝑥1 = ?
𝑨𝟐 75 𝒙𝟐 15 𝒚𝟐 12.5 𝑦2 = ? 𝑥2 = ?
𝑨𝟑 100 𝒙𝟑 15 𝒚𝟑 22.5 𝑦3 = ? 𝑥3 = ?
Q. Determine the MOI of I section
beam about its centroidal axis

33. 𝑰𝒙𝒙𝟐 = 𝑰𝒄𝒙𝟐
+ 𝑨𝟐(𝒚)𝟐
𝑰𝒙𝒙𝟑 = 𝑰𝒄𝒙𝟑
+ 𝑨𝟑(𝒚)𝟐
b
d
𝑰𝒄𝒙 𝑰𝒄𝒚
33
𝑰𝒙𝒙𝟏 = 𝑰𝒄𝒙𝟏
+ 𝑨𝟏(𝒚)𝟐

34. 𝑰𝒚𝒚𝟏 = 𝑰𝒄𝒚𝟏
+ 𝑨𝟏(𝒙)𝟐
𝑰𝒚𝒚𝟐 = 𝑰𝒄𝒚𝟐
+ 𝑨𝟐(𝒙)𝟐
𝑰𝒚𝒚𝟑 = 𝑰𝒄𝒚𝟑
+ 𝑨𝟑(𝒙)𝟐
34

35. Q. Determine the MOI of Z
section beam about its
centroidal axis.
35
Area Value
𝒄𝒎𝟐
X-
coord
inate
Value
(cm)
Y-
coord
inate
Value
(cm)
𝒚(cm)
𝒚𝒏 − 𝒚𝒄
𝒙 (cm)
𝒙𝒏 − 𝒙𝒄
𝑨𝟏 100 𝒙𝟏 17.5 𝒚𝟏 2.5 𝑦1 = ? 𝑥1 = ?
𝑨𝟐 37.5 𝒙𝟐 8.75 𝒚𝟐 12.5 𝑦2 = ? 𝑥2 = ?
𝑨𝟑 25 𝒙𝟑 5 𝒚𝟑 21.25 𝑦3 = ? 𝑥3 = ?

36. Q. Determine the MOI of Z
section beam about its
centroidal axis.
36
Area Value
𝒄𝒎𝟐
X-
coord
inate
Value
(cm)
Y-
coord
inate
Value
(cm)
𝒚(cm)
𝒚𝒏 − 𝒚𝒄
𝒙 (cm)
𝒙𝒏 − 𝒙𝒄
𝑨𝟏 100 𝒙𝟏 17.5 𝒚𝟏 2.5 𝑦1 = ? 𝑥1 = ?
𝑨𝟐 37.5 𝒙𝟐 8.75 𝒚𝟐 12.5 𝑦2 = ? 𝑥2 = ?
𝑨𝟑 25 𝒙𝟑 5 𝒚𝟑 21.25 𝑦3 = ? 𝑥3 = ?
𝒄𝟏 𝟏𝟕. 𝟓, 𝟐. 𝟓
𝒄𝟐 𝟖. 𝟕𝟓, 𝟏𝟐. 𝟓
𝒄𝟑 𝟓, 𝟐𝟏. 𝟐𝟓
𝑪 𝟏𝟑. 𝟓𝟓, 𝟕. 𝟔𝟗

37. 𝑰𝒙𝒙𝟏 = 𝑰𝒄𝒙𝟏
+ 𝑨𝟏(𝒚)𝟐
𝑰𝒙𝒙𝟐 = 𝑰𝒄𝒙𝟐
+ 𝑨𝟐(𝒚)𝟐
𝑰𝒙𝒙𝟑 = 𝑰𝒄𝒙𝟑
+ 𝑨𝟑(𝒚)𝟐
37

38. 𝑰𝒚𝒚𝟏 = 𝑰𝒄𝒚𝟏
+ 𝑨𝟏(𝒙)𝟐
𝑰𝒚𝒚𝟐 = 𝑰𝒄𝒚𝟐
+ 𝑨𝟐(𝒙)𝟐
𝑰𝒚𝒚𝟑 = 𝑰𝒄𝒚𝟑
+ 𝑨𝟑(𝒙)𝟐
38

39. Determine the MOI of beam about its centroidal X’ axis.
(Hibler page # 789)
39

40. Eg. A-3, Determine the MOI of beam about its centroidal X &
Y axis. (Hibler page # 790)
40

41. 41

42. 42

43. 43

44. 44

45. 45

46. 46