9.3 Composite Bodies Consists of a series of connected “simpler” shaped bodies, which may be rectangular, triangular or semicircular A body can be sectioned

9.3 Composite Bodies9.3 Composite Bodies

Consists of a series of connected “simpler” shaped bodies, which may be rectangular, triangular or semicircular

A body can be sectioned or divided into its composite parts

Provided the weight and location of the center of gravity of each of these parts are known, the need for integration to determine the center of gravity for the entire body can be neglected


Accounting for finite number of weights

Where represent the coordinates of the

center of gravity G of the composite bodyrepresent the coordinates of the

center of gravity at each composite part of the body

represent the sum of the weights of all the composite parts of the body or total weight

W

Wzz

W

Wyy

W

Wxx

~~~

W

zyx

zyx

~,~,~

,,


When the body has a constant density or specified weight, the center of gravity coincides with the centroid of the body

The centroid for composite lines, areas, and volumes can be found using the equation

However, the W’s are replaced by L’s, A’s and V’s respectively

WWz

zWWy

yWWx

x

~~~


Procedure for AnalysisComposite Parts Using a sketch, divide the body or

object into a finite number of composite parts that have simpler shapes

If a composite part has a hole, or a geometric region having no material, consider it without the hole and treat the hole as an additional composite part having negative weight or size


Procedure for AnalysisMoment ArmsEstablish the coordinate axes on

the sketch and determine the coordinates of the center of gravity or centroid of each part


Procedure for AnalysisSummationsDetermine the coordinates of the

center of gravity by applying the center of gravity equations

If an object is symmetrical about an axis, the centroid of the objects lies on the axis


Example 9.9Locate the centroid of the wire.

SolutionComposite Parts

Moment Arms Location of the centroid for each piece is

determined and indicated in the diagram



SolutionSummations

Segment

L (mm)

x (mm) y (mm) z (mm) xL (mm2)

yL (mm2)

zL (mm2)

1 188.5 60 -38.2 0 11 310 -7200 0

2 40 0 20 0 0 800 0

3 20 0 40 -10 0 800 -200

Sum 248.5 11 310 -5600 -200


SolutionSummations

mmL

Lzz

mmL

Lyy

mmL

Lxx

805.05.248

200~

5.225.248

5600~

5.455.248

11310~


Example 9.10Locate the centroid of the plate

area.


SolutionComposite Parts Plate divided into 3 segments Area of small rectangle considered

“negative”


SolutionMoment Arm Location of the centroid for each piece is



SolutionSummationsSegment A (mm2) x (mm) y (mm) xA (mm3) yA (mm3)

1 4.5 1 1 4.5 4.5

2 9 -1.5 1.5 -13.5 13.5

3 -2 -2.5 2 5 -4

Sum 11.5 -4 14


SolutionSummations

mmA

Ayy

mmA

Axx

22.15.11

14~

348.05.11

4~


Example 9.11Locate the center of mass of the composite assembly. The conical frustum has a density of ρc = 8Mg/m3 and the hemisphere

has a density of ρh = 4Mg/m3.

There is a 25mm radius cylindrical hole in the center.


SolutionComposite PartsAssembly divided into 4 segmentsArea of 3 and 4 considered

“negative”

View Free Body Diagram


SolutionMoment Arm Location of the centroid for each piece is


Summations Because of symmetry,

0yx


SolutionSummationsSegment m (kg) z (mm) zm (kg.mm)

1 4.189 50 209.440

2 1.047 -18.75 -19.635

3 -0.524 125 -65.450

4 -1.571 50 -78.540

Sum 3.141 45.815


SolutionSummations

mmm

mzz 6.14

141.3

815.45~

9.4 Theorems of Pappus and Guldinus


A surface area of revolution is generated by revolving a plane curve about a non-intersecting fixed axis in the plane of the curve

A volume of revolution is generated by revolving a plane area bout a nonintersecting fixed axis in the plane of area

Example Line AB is rotated about

fixed axis, it generates the surface area of a cone (less area of base)



Example Triangular area ABC rotated

about the axis would generate the volume of the cone

The theorems of Pappus and Guldinus are used to find the surfaces area and volume of any object of revolution provided the generating curves and areas do not cross the axis they are rotated



Surface Area Area of a surface of revolution = product of

length of the curve and distance traveled by the centroid in generating the surface area LrA



Volume Volume of a body of revolution = product

of generating area and distance traveled by the centroid in generating the volume

ArV



Composite Shapes The above two mentioned theorems

can be applied to lines or areas that may be composed of a series of composite parts

Total surface area or volume generated is the addition of the surface areas or volumes generated by each of the composite parts

ArV

LrA~

~



Example 9.12Show that the surface area of a sphere is

A = 4πR2 and its volume

V = 4/3 πR3



SolutionSurface Area Generated by rotating semi-arc about

the x axis For centroid,

For surface area,

242

2

;~

/2

RRR

A

LrA

Rr

View Free Body Diagram



SolutionVolume Generated by rotating semicircular area

about the x axis For centroid,

For volume,

32

34

21

34

2

;~

3/4

RRR

V

ArV

Rr

9.5 Resultant of a General Distributed

Loading


LoadingPressure Distribution over a Surface Consider the flat plate subjected to the

loading function ρ = ρ(x, y) Pa Determine the force dF acting on the

differential area dA m2 of the plate, located at the differential point (x, y)dF = [ρ(x, y) N/m2](d A m2)

= [ρ(x, y) d A]N Entire loading represented as

infinite parallel forces acting on separate differential area dA


Loading


LoadingPressure Distribution over a SurfaceThis system will be simplified to a

single resultant force FR acting through a unique point on the plate

Pressure Distribution over a SurfaceMagnitude of Resultant Force To determine magnitude of FR, sum the

differential forces dF acting over the plate’s entire surface area dA

Magnitude of resultantforce = total volume underthe distributed loadingdiagram


Loading


Loading

A VR dVdAyxFF ),(;


Loading


LoadingPressure Distribution over a SurfaceLocation of Resultant Force

Line of action of action of the resultant force passes through the geometric center or centroid of the volume under the distributed loading diagram

V

V

A

A

V

V

A

A

dV

ydV

dAyx

dAyxyy

dV

xdV

dAyx

dAyxxx

),(

),(

),(

),(

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9.3 Composite Bodies Consists of a series of connected “simpler” shaped bodies, which may be rectangular, triangular or semicircular A body can be sectioned