Center of gravity and Centroids Part 2 MET 2214 Statics (MET 2214) Prof. Simin Nasseri Composite Bodies Objective: You will be able to determine: The location of the center of gravity, the center of mass, and centroid using the method of composite bodies. Statics (MET 2214) Prof. Simin Nasseri APPLICATIONS The I-beam is commonly used in building structures. When doing a stress analysis on an I - beam, the location of the centroid is very important. How can we easily determine the location of the centroid for a given beam shape? Statics (MET 2214) Prof. Simin Nasseri

APPLICATIONS Cars, trucks, bikes, etc., are assembled using many individual components. When designing for stability on the road, it is important to know the location of the bikes center of gravity (CG). If we know the weight and CG of individual components, how can we determine the location of the CG of the assembled unit? Statics (MET 2214) Prof. Simin Nasseri CONCEPT OF A COMPOSITE BODY a e b a b e d

d Many industrial objects can be considered as composite bodies made up of a series of connected simpler shaped parts or holes, like a rectangle, triangle, and semicircle. Knowing the location of the centroid, C, or center of gravity, G, of the simpler shaped parts, we can easily determine the location of the C or G for the more complex composite body. Statics (MET 2214) Prof. Simin Nasseri CONCEPT OF A COMPOSITE BODY a e b a b e d d This can be done by considering each part as a particle and following the procedure as described later.

This is a simple, effective, and practical method of determining the location of the centroid or center of gravity. Statics (MET 2214) Prof. Simin Nasseri 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 weight, mass, or size (depending on the problem), 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 the center of gravity 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 = ( xi Ai ) / ( Ai ) or x = ( xi Wi ) / ( Wi ) This approach will become clear by doing examples! Statics (MET 2214) Prof. Simin Nasseri Centroids of common shapes y y, y'

r C y y' x' x C P r ( 2 ) Statics (MET 2214) Prof. Simin Nasseri h x' r x 0 4r y 3 A 12 r 2 y' c

x 4r 3 A 14 r 2 x y P r ( 2 12 ) x' C x b x 13 (b c ) y 13 h A 12 bh EXAMPLE Given: The part shown. Find: The centroid of the part. a c b

d Solution: 1. This body can be divided into the following pieces: rectangle (a) + triangle (b) + quarter circular (c) semicircular area (d) Statics (MET 2214) Prof. Simin Nasseri EXAMPLE Steps 2 & 3: Make up and fill the table using parts a, c b, c, and d. a b d Segment Rectangle Triangle Q. Circle Semi-Circle

Statics (MET 2214) Prof. Simin Nasseri Area A (in2) x y (in) 18 4.5 9/4 /2 3 7 4(3) / (3 ) 0 28.0 (in)

Ax ( in3) Ay ( in3) 1.5 1 4(3) / (3 ) 4(1) / (3 ) 54 31.5 9 0 27 4.5 9 - 2/3 76.5 39.83 EXAMPLE C 4. Now use the table data and these formulas to find the

coordinates of the centroid. x = ( xA) / ( A ) = 76.5 in3/ 28.0 in2 = 2.73 in y = ( y A) / ( A ) = 39.83 in3 / 28.0 in2 = 1.42 in Statics (MET 2214) Prof. Simin Nasseri CONCEPT QUIZ 1. Based on the typical centroid information available in handbooks, what are the minimum number of segments you will have to consider for determing the centroid of the given area? 1, 2, 3, or 4 Statics (MET 2214) Prof. Simin Nasseri 3cm 1 cm 1 cm 3cm CONCEPT QUIZ 2.

A storage box is tilted up to clean the rug underneath the box. It is tilted up by pulling the handle C, with edge A remaining on the ground. What is the maximum angle of tilt (measured between bottom AB and the ground) possible before the box tips over? A) 30 B) 45 C) 60 C G D) 90 B 30 B G 30 60 A

Statics (MET 2214) Prof. Simin Nasseri W should pass through A in order to have no moment (about A), otherwise the box tips over. A EXAMPLE Given: Two blocks of different materials are assembled as shown. The densities of the materials are: A = 150 lb / ft3 and B = 400 lb / ft3. Find: The center of gravity of this assembly. Solution 1. In this problem, the blocks A and B can be considered as two segments. Statics (MET 2214) Prof. Simin Nasseri

EXAMPLE Weight = w = (Volume in ft3) wA = 150 (0.5) (6) (6) (2) / (12)3 = 3.125 lb wB = 400 (6) (6) (2) / (12)3 = 16.67 lb Segment w (lb) x (in) y (in) z (in) w x(lbin) w y (lbin) wz (lbin)

A B 3.125 16.67 4 1 1 3 2 3 12.5 16.67 3.125 50.00 6.25 50.00 19.79 29.17 53.12

56.25 Statics (MET 2214) Prof. Simin Nasseri EXAMPLE ~ x = ( x w) / ( w ) = 29.17/19.79 = 1.47 in ~ y = ( y w) / ( w ) = 53.12/ 19.79 = 2.68 in ~ z = ( z w) / ( w ) = 56.25 / 19.79 = 2.84 in Statics (MET 2214) Prof. Simin Nasseri READING QUIZ 1. A composite body in this section refers to a body made of ____. A) carbon fibers and an epoxy matrix B) steel and concrete 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) integration B) differentiation

C) simple arithmetic D) All of the above. Statics (MET 2214) Prof. Simin Nasseri ATTENTION QUIZ y 1. 2. 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 A) (1, 1) m B) (1.25, 1.25) m C) (0.5, 0.5 ) m

D) (1.5, 1.5) m 2cm 2 4cm 3 A2-A1-A3 For determining the centroid of the area, two square segments are considered; square ABCD and square DEFG. What are ~ ~ (x, y ) of the centroid of the coordinates square DEFG? Statics (MET 2214) Prof. Simin Nasseri 1 x 2cm y A 2cm

1m 1m D E 1m F B G 1m C x