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ENGR-1100 Introduction to Engineering
Lecture 17
Analysis
CENTROID OF COMPOSITE AREAS
In-Class Activities :
Today’s Objective :
Students will:
a) Understand the concept of centroid.
- Applications
- Centroid
- Determine Centroid Location
•Method of Composite
a) Understand the concept of centroid.
b) Be able to determine the location of the centroid using the method of composite areas.
Areas
- Concept Quiz
- Group Problem Solving
- Attention Quiz
CENTROID OF A BODY
= =
y~dA y
~xdA x
Similarly, the coordinates of the centroid of volume, area, or
length can be obtained by replacing W by V, A, or L,
respectively.
= = dA
y dA
x
CONCEPT OF CENTROID
The centroid coincides with the center of
The centroid, C, is a point defining the
geometric center of an object.
The centroid coincides with the center of
mass or the center of gravity only if the
material of the body is homogenous (density
or specific weight is constant throughout the
body).
If an object has an axis of symmetry, then
ththe centroid of object lies on that axis. t id f bj t li th t i
In some cases, the centroid may not be
located on the object.
EXAMPLE (continued)
4. x = ( A ~x dA ) / ( A dA )
x (x 3 ) d x 1/5 [ x 5 ]^1
1
0 ^ x (x ) d x^ 1/5 [ x ]
0 ^ (x^
3 ) d x 1/4 [ x 4 ] 1
= ( 1/5) / ( 1/4) = 0.8 m
11
A y dA 0 (x 3 / 2) ( x 3 ) dx 1/14[x 7 ]^1
A dA 0 x 3 dx 1/
y = = 1
= (1/14) / (1/4) = 0.2857 m
APPLICATIONS
The I-beam (top) or T-beam
(bottom) shown are commonly
used in building various typesused in building various types
of structures.
When doing a stress or
deflection analysis for a beam,
the location of its centroid is
very important.
How can we easily determine
the location of the centroid for
different beam shapes?
STEPS FOR ANALYSIS
- Divide the body into pieces that are known shapes. Holes are considered as pieces with negative weight or size.
2 2. M kMake a table with the first column for segment number, the second t bl ith th fi t l f t b th d column for size, the next set of columns for the moment arms, and, finally, several columns for recording results of simple intermediate calculations.
- Fix the coordinate axes, determine the coordinates of centroid of each piece, and then fill in the table.
- Sum the columns to get x, y, and z. Use formulas like
x = ( Σ x (^) i Ai ) / ( Σ Ai ) This approach will become straightforward by doing examples!
EXAMPLE
Given: The part shown.
Find: The centroid of
h
Solution :
1. This body can be divided into the following pieces:
the part.
Plan: Follow the steps
for analysis.
rectangle (a) + triangle (b) + quarter circular (c) –
semicircular area (d). Note the negative sign on the hole!
READING QUIZ
1. A composite body in this section refers to a body made of ____.
A) Carbon fibers and an epoxy matrix in a car fender
B) Steel and concrete forming a structure
C) A collection of “simple” shaped parts or holes
D) A collection of “complex” shaped parts or holes
2. The composite method for determining the location of the
center of gravity of a composite body requires _______.
A) Simple arithmetic B) Integration
C) Differentiation D) All of the above.
CONCEPT QUIZ
Based on the typical centroid information, what are the minimum number of pieces you will have to id f d i i h id f
3cm 1 cm
1 cm
consider for determining the centroid of the area shown at the right? A) 4 B) 3 C) 2 D) 1
3cm
ATTENTION QUIZ
- A rectangular area has semicircular and triangular cuts as shown. For determining the centroid, what is the minimum number of
2cm
4cm
y
- For determining the centroid of the area, two square segments are considered; square ABCD
pieces that you can use? A) Two B) Three
C) Four D) Five
2cm 2cm
4cm
x
A
y 1m (^) 1m
and square DEFG. What are the coordinates D (x, y ) of the centroid of square DEFG? A) (1, 1) m B) (1. 25, 1. 25) m
C) (0. 5, 0. 5 ) m D) (1.5, 1.5) m
A
1m
1m
E
F G
B (^) C x
D
GROUP PROBLEM SOLVING
Given: A plate as shown.
Find: The location of its centroid
Plan:
Follow the solution steps to
find the centroid by integration.
