Centroid of Composite Areas

ENGR-1100 Introduction to Engineering Analysis

Lecture 17 CENTROID OF COMPOSITE AREAS

Today’s Objective : In-Class Activities: Students will: • Reading Quiz a) Understand the concept of centroid. b) Be able to determine the location of • Applications the centroid using the method of • Centroid composite areas. • Determine Centroid Location •Method of Composite Areas • Concept Quiz • Group Problem Solving • Attention Quiz 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 axis. In some cases, the centroid may not be located on the object. CENTROID OF A BODY

∫ ~xdA ∫ ~ydA x = y = ∫ dA ∫ dA STEPS TO DETERMINE THE CENTROID OF AN AREA 1. 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 = x2 + 1), 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. STEPS TO DETERMINE THE CENTROID OF AN AREA

The full list is on LMS! EXAMPLE Given: The area as shown. Find: The centroid location (x , y) Plan: Follow the steps.

Solution: 1. Since y is given in terms of x, choose dA as a vertical rectangular strip.

2. dA = y dx = x3 dx ~ ~ 3. x = x and y = y / 2 = x3 / 2 EXAMPLE(continued)

~ 4. x = ( ∫A x dA ) / ( ∫A dA )

1 3 5 1 0∫ x (x ) d x 1/5 [ x ] 0 = = ∫1 (x3 ) d x 1/4 [ x4 ]1 0 0 = ( 1/5) / ( 1/4) = 0.8 m

1 ~ 3 3 7 1 ∫A y dA 0 ∫ (x / 2) ( x ) dx 1/14[x ] y = = = 0 1 3 ∫A dA 0 ∫ x dx 1/4 = (1/14) / (1/4) = 0.2857 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? STEPS FOR ANALYSIS

1. 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 xi Ai ) / ( Σ Ai ) This approach will become straightforward by doing examples! EXAMPLE

Given: The part shown. Find: The centroid of the part. Plan: Follow the steps for analysis. Solution: 1. 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! 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 Area A x y x A y A (in2) (in) (in) ( in3) ( in3) Rectangle 18 3 1.5 54 27 Triangle 4.5 7 1 31.5 4.5 Q. Circle 9 π / 4 – 4(3) / (3 π) 4(3) / (3 π) – 9 9 Semi-Circle – π / 2 0 4(1) / (3 π) 0 - 2/3

Σ 28.0 76.5 39.83 EXAMPLE (continued)

4. Now use the table data results and the formulas to find the coordinates of the centroid. ∼ ∼ Area A x A y A 28.0 76.5 39.83 x = ( Σ ∼x A) / ( Σ A ) = 76.5 in3/ 28.0 in2 = 2.73 in y = ( Σ ∼y A) / (Σ A ) = 39.83 in3 / 28.0 in2 = 1.42 in 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 3cm 1 cm Based on the typical centroid information, what are the minimum 1 cm number of pieces you will have to consider for determining the centroid of the area shown at the right? 3cm A) 4 B) 3 C) 2 D) 1 ATTENTION QUIZ y 1. A rectangular area has semicircular and 2cm triangular cuts as shown. For determining the centroid, what is the minimum number of 4cm pieces that you can use? A) Two B) Three x 2cm 2cm C) Four D) Five

2. For determining the centroid of the area, two y 1m 1m square segments are considered; square ABCD and square DEFG. What are the coordinates A D ~ ~ E 1m (x, y ) of the centroid of square DEFG? G F A) (1, 1) m B) (1.25, 1.25) m 1m B x C) (0.5, 0.5 ) m D) (1.5, 1.5) m C 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. (1) Define dA in the blue area; (2) Find x- and y-centroid in dA; (3) Integrate dA in the domain to find the total area; (4) Use integration to find x- ∫ ~xdA ∫ ~ydA and y-centroids for the x = y = entire shape (left equations). ∫ dA ∫ dA GROUP PROBLEM SOLVING (continued)

Solution 1. Choose dA as a vertical rectangular strip. GROUP PROBLEM SOLVING (continued)