Centroid And Center of Gravity
Centroid And
Center of Gravity
|
Definition Of
Centroid:
In other words, the centroid is a point of plane lamina or figure at which weight of whole lamina is to be act.
OR
The center of mass of a geometric object of uniform density is called as Centroid.
Definition of
center of gravity:
The center of
gravity is a point in body at which weight of whole body is to be act.
Difference
between centroid and center of gravity:
The major
difference between centroid and center of gravity is, centroid is term related
with plane lamina or figure or 2dimensional objects while center of gravity is
term which related with 3dimensional bodies.
In civil engineering
it is very important while designing structure to find the center of gravity.
While designing combined footing centroid is calculated to find position of
resultant force. In retaining wall structure resultant of forces is acting on
center of gravity. Also in gravity dam and earthen dam it is very necessary to
find the center of gravity.
Moment of an
area about an axis: (Varigons theorem)
Stating that the moment of area
equals the summation of area times distance to an axis [Σ(a × d)].
Varigons theorem:
The theorem
states that the moment of a resultant of two concurrent forces about any
point is equal to the algebraic sum of the moments of its components
about the same point.
In simple words if whole fig. or body is divided
in various parts then summation of moments of areas of an individual fig. or
body is equals to moment of area of whole fig. i.e. in following fig. of
I-Section whole fig. is divided in 3 parts.
Part 1 is top
flange of I-section. Part 2 is vertical web of I-section. Part 3 is bottom
flange of I-section. Origin of X-axis and Y-axis (0,0) is at the bottom left of
I-section. i.e. this fig. lies in 1st quadrant. With respective to this axis centroidal distance find out.
Centroidal
distance of each individual fig. is find out. with respective X-axis and
Y-axis. i.e. for fig.1 X1 is cetroidal distance from Y-axis and Y1 is cetroidal
distance from X-axis. Similarly X2, Y2 and X3, Y3 is centroidal distance of
fig.2 and fig.3 with respect to X-axis and Y-axis respectively.
Finding
centroidal distance x̄ :
A1, A2 and A3 are the areas of fig. 1,
2, and 3 respectively.
A1X1 is moment of area about Y-axis of
fig.1. Similarly A2X2, and A3X3 are moment of areas about Y-axis of fig.2 and
fig.3.
A x̄ is moment of area of whole area about
Y-axis.
Now, according
to varigons theorem summation of all moments of areas of individual fig. i.e.
A1X1+A2X2+A3X3 is equals to moments of areas of total fig. i.e. A x̄.
A x̄ =
A1X1+A2X2+A3X3
Simmilarly,
Finding
centroidal distance Ȳ :
A1, A2 and A3 are the areas of fig. 1,
2, and 3 respectively.
A1Y1 is moment of area about X-axis of
fig.1. Similarly A2Y2, and A3Y3 are moment of areas about X-axis of fig.2 and
fig.3.
A is moment of area of whole area
about X-axis.
Now, according to varigons theorem
summation of all moments of areas of individual fig. i.e. A1Y1+A2Y2+A3Y3 is
equals to moments of areas of total fig. i.e. AȲ.
AȲ = A1Y1+A2Y2+A3Y3
Procedure
to finding Centroid:
Step1:
Divide total area in to basic geometric fig.
Step2:
Calculate areas basic geometric fig.
Step3:
Calculate centroidal distance with respect to X-axis
and Y-axis (X1, Y1, X2, Y2,……. Xn,Yn.)
Step4:
Calculate centroidal distance by varigon’s theorem.
Prob.1: Locate
the centroid of angle section 90 mm × 100 mm × 10 m. (90 mm side is vertical)
1.
Dividing given area in basic
areas. As shown.
2.
Calculating Areas:
A1
= Area of fig. 1. = 10 X 90 = 900mm2.
A2
= Area of fig. 2. = 90 X 10 = 900mm2.
1.
Finding centroidal distance of
basic areas.
X1
= Centroidal distance of area 1 along X-axis or Centroidal distance of area 1 from
Y-axis = 10/2 = 5 mm.
X2
= Centroidal distance of area 2 along X-axis or Centroidal distance of area 2 from
Y-axis = 10 + (90/2) = 55 mm.
(Note:
Here we calculating centroidal distance for area 2 by calculating distance
between centroidal point of area 2 and Y- axis. i.e. 10 + (90/2)=55mm)
Y1 = Centroidal distance of area
1 along Y-axis or Centroidal distance of area 1 from X-axis = 90/2 = 45 mm.
Y2
= Centroidal distance of area 2 along Y-axis or Centroidal distance of area 2 from
X-axis = 10/2 = 5mm.
1.
Calculating centroidal distance x̄ and Ȳ:
Prob.2:
Find the centroid of an inverted T-section from the bottom, if flange is 60 cm
× 10 cm and web is 10 cm × 60 cm.
1.
Dividing given area in basic areas.
As shown.
2.
Calculating Areas:
A1
= Area of fig. 1. = 60 X 10 = 600cm2.
A2
= Area of fig. 2. = 10 X 60 = 600cm2.
A
= Area of total fig. = A1 + A2 = 1200cm2.
3.
Finding centroidal distance of
basic areas.
X1
= Centroidal distance of area 1 along X-axis or Centroidal distance of area 1 from
Y-axis = 60/2 = 30 cm.
X2
= Centroidal distance of area 2 along X-axis or Centroidal distance of area 2 from
Y-axis = 60/2 = 30 cm.
(Note:
Here section is symmetrical along Y-axis hence centroidal distance along X
direction of all areas are same. All X1 and X2 values are same
i.e.x̄ =30cm)
Y1
= Centroidal distance of area 1 along Y-axis or Centroidal distance of area 1 from
X-axis = 10/2 = 5 cm.
Y2
= Centroidal distance of area 2 along Y-axis or Centroidal distance of area 2 from
X-axis = 10 + (60/2) = 40 cm.
(Note:
Here we calculating centroidal distance for area 2 by calculating distance
between centroidal point of area 2 and X- axis. i.e. (60/2)+10=40cm)
4.
Calculating centroidal distance x̄ and Ȳ:
1 comments
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