Lecture 17. ENGR-1100 Introduction to Engineering Analysis CENTROID OF COMPOSITE AREAS

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b) Be able to determine the location of the centroid using the method of composite areas.

1 ENGR-00 Introduction to Engineering Analysis Lecture 7 CENTROID OF COMPOSITE AREAS Today s Objective : Students will: a) Understand the concept of centroid. b) Be able to determine the location of the centroid using the method of composite areas. In-Class Activities: Reading Quiz Applications Centroid Determine Centroid Location Method of Composite Areas Concept Quiz Group Problem Solving Attention Quiz

CENTROID OF A BODY x~ da x = da y~ da y = da Similarly, the coordinates of the centroid of volume, area, or length can be obtained by replacing W by V, A, or L, respectively.

2 CENTROID OF A BODY x~ da x = da y~ da y = da Similarly, the coordinates of the centroid of volume, area, or length can be obtained by replacing W by V, A, or L, respectively. CONCEPT OF CENTROID 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 the centroid of object lies on that t axis. In some cases, the centroid may not be located on the object. 2

STEPS TO DETERME THE CENTROID OF AN AREA. Choose an appropriate differential element da at a general point (x,y). Hint: Generally, if y is easily expressed in terms of x (e.g., y = x 2 + ), use a vertical rectangular element.

Determine coordinates (x, y) of the centroid of the rectangular element in terms of the general point (x,y). 4.

3 STEPS TO DETERME THE CENTROID OF AN AREA. Choose an appropriate differential element da at a general point (x,y). Hint: Generally, if y is easily expressed in terms of x (e.g., y = x 2 + ), use a vertical rectangular element. If the converse is true, then use a horizontal rectangular element. 2. Express da in terms of the differentiating element dx (or dy). ~ ~ 3. Determine coordinates (x, y) of the centroid of the rectangular element in terms of the general point (x,y). 4. Express all the variables and integral limits in the formula using either x or y depending on whether the differential element is in terms of dx or dy, respectively, and integrate. EXAMPLE Given: The area as shown. Find: The centroid location (x, y) Plan: Fll Follow the steps. Solution:. Since y is given in terms of x, choose da as a vertical rectangular strip. 2. da = y dx = x 3 dx ~ ~ 3. x = x and y = y / 2 = x 3 / 2 3

4. x = ( ~ A x da ) / ( A da ) EXAMPLE(continued) 0 x(x 3 )dx /5 [ x 5 ] = = 0 (x 3 ) d

2857 m APPLICATIONS The I-beam (top) or T-beam (bottom) shown are commonly used in

4 4. x = ( ~ A x da ) / ( A da ) EXAMPLE(continued) 0 x(x 3 )dx /5 [ x 5 ] = = 0 (x 3 ) d x /4 [ x 4 ] = ( /5) / ( /4) = 0.8 m 0 0 ~ A y da 0 (x 3 / 2) ( x 3 ) dx /4[x 7 ] 0 y = = = A da 0 x 3 dx /4 = (/4) / (/4) = m APPLICATIONS The I-beam (top) or T-beam (bottom) shown are commonly used 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? 4

5 STEPS FOR ANALYSIS. Divide the body into pieces that are known shapes. Holes are considered as pieces with negative weight or size. 2. Make a table with the first column for segment number, the second column for size, the next set of columns for the moment arms, and, finally, several columns for recording results of simple intermediate calculations. 3. Fix the coordinate axes, determine the coordinates of centroid of each piece, and then fill in the table. 4. Sum the columns to get x, y, and z. Use formulas like x = ( Σ x i A i ) / ( Σ A i ) This approach will become straightforward by doing examples! Solution: EXAMPLE Given: The part shown. Find: The centroid of the part. Plan: Follow the steps for analysis.. This body can be divided into the following pieces: rectangle (a) + triangle (b) + quarter circular (c) semicircular area (d). Note the negative sign on the hole! 5

EXAMPLE (continued) Steps 2 & 3: Make up and fill the table using parts a, b, c, and d. Note the location of the axis system. Segment Rectangle Triangle Q. Circle Semi-Circle Area A (in 2 ) 8 4.

6 EXAMPLE (continued) Steps 2 & 3: Make up and fill the table using parts a, b, c, and d. Note the location of the axis system. Segment Rectangle Triangle Q. Circle Semi-Circle Area A (in 2 ) π / 4 π / 2 x y x A y A (in) (in) ( in 3 ) ( in 3 ) 3 7 4(3) / (3 π) ).5 4(3) / (3 π) ) () / (3 π) 0-2/3 Σ EXAMPLE (continued) C 4. Now use the table data results and the formulas to find the coordinates of the centroid. Area A x A y A x = ( Σ x A) / ( Σ A ) = 76.5 in 3 / 28.0 in 2 = 2.73 in y = ( Σ y A) / (Σ A ) = in 3 / 28.0 in 2 =.42 in 6

7 READING QUIZ. 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 consider for determining i the centroid of the area shown at the right? A) 4 B) 3 C) 2 D) 3cm cm cm 3cm 7

For determining the centroid of the area, two square segments are considered; square ABCD and square DEFG.

8 ATTENTION QUIZ y. A rectangular area has semicircular and triangular cuts as shown. For determining the centroid, what is the minimum number of pieces that you can use? A) Two B) Three C) Four D) Five 2cm 2cm 2cm 4cm x 2. For determining the centroid of the area, two square segments are considered; square ABCD and square DEFG. What are the coordinates ~~ (x, y ) of the centroid of square DEFG? A) (, ) m B) (.25,.25) m C) (0.5, 0.5 ) m D) (.5,.5) m A y m m B D E G F C m m x 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. 8

9 GROUP PROBLEM SOLVING (continued) Solution. Choose da as a vertical rectangular strip. GROUP PROBLEM SOLVING (continued) 9

QUIZ 4 (CHAPTER 17) SOLUTIONS MATH 252 FALL 2008 KUNIYUKI SCORED OUT OF 125 POINTS MULTIPLIED BY % POSSIBLE

QUIZ 4 (CHAPTER 17) SOLUTIONS MATH 5 FALL 8 KUNIYUKI SCORED OUT OF 15 POINTS MULTIPLIED BY.84 15% POSSIBLE 1) Reverse the order of integration, and evaluate the resulting double integral: 16 y dx dy. Give