izmir institute of technology department of architecture ar231 fall12/13 15.01.2015 dr. engin aktaş...

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 1

Distributed Forces: Distributed Forces: Centroids and Centers of GravityCentroids and Centers of Gravity

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 2

Introduction

• The earth exerts a gravitational force on each of the particles forming a body. All these forces can be represented by a single equivalent force equal to the weight of the body and applied at the center of gravity for the body.

• The centroid of an area is similar to the center of gravity of a body. The concept of the first moment of an area is used to locate the center of gravity.

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 3

Center of Gravity of a 2D Body• Center of gravity of a plate

dWy

WyWyM

dWx

WxWxM

x

y

• Center of gravity of a wire

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 4

Centroids and First Moments of Areas and Lines

x

QdAyAy

y

QdAxAx

dAtxAtx

dWxWx

x

y

respect toh moment witfirst

respect toh moment witfirst

• Centroid of an area

dLyLy

dLxLx

dLaxLax

dWxWx

• Centroid of a line

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 5

First Moments of Areas and Lines• An area is symmetric with respect to an axis BB’

if for every point P there exists a point P’ such that PP’ is perpendicular to BB’ and is divided into two equal parts by BB’.

• The first moment of an area with respect to a line of symmetry is zero.

• If an area possesses a line of symmetry, its centroid lies on that axis

• If an area possesses two lines of symmetry, its centroid lies at their intersection.

• An area is symmetric with respect to a center O if for every element dA at (x,y) there exists an area dA’ of equal area at (-x,-y).

• The centroid of the area coincides with the center of symmetry.

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 6

Centroids of Common Shapes of Areas

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 7

Centroids of Common Shapes of Lines

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 8

Composite Plates and Areas• Composite plates

WyWY

WxWX

• Composite area

AyAY

AxAX

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 9

Sample Problem 5.1

For the plane area shown, determine the first moments with respect to the x and y axes and the location of the centroid.

SOLUTION:

• Divide the area into a triangle, rectangle, and semicircle with a circular cutout.

• Compute the coordinates of the area centroid by dividing the first moments by the total area.

• Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout.

• Calculate the first moments of each area with respect to the axes.

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 10

Sample Problem 5.1

33

33

mm10758

mm10506

y

x

Q

Q• Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout.

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 11

Sample Problem 5.1

23

33

mm1013.828

mm107.757

A

AxX

mm 8.54X

23

33

mm1013.828

mm102.506

A

AyY

mm 6.36Y

• Compute the coordinates of the area centroid by dividing the first moments by the total area.

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 12

Determination of Centroids by Integration

ydxy

dAyAy

ydxx

dAxAx

el

el

2

dyxay

dAyAy

dyxaxa

dAxAx

el

el

2

drr

dAyAy

drr

dAxAx

el

el

2

2

2

1sin

3

2

2

1cos

3

2

dAydydxydAyAy

dAxdydxxdAxAx

el

el• Double integration to find the first moment

may be avoided by defining dA as a thin rectangle or strip.

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 13

Sample Problem 5.4

Determine by direct integration the location of the centroid of a parabolic spandrel.

SOLUTION:

• Determine the constant k.

• Evaluate the total area.

• Using either vertical or horizontal strips, perform a single integration to find the first moments.

• Evaluate the centroid coordinates.

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 14

Sample Problem 5.4SOLUTION:

• Determine the constant k.

2121

22

22

2

yb

axorx

a

by

a

bkakb

xky

• Evaluate the total area.

3

30

3

20

22

ab

x

a

bdxx

a

bdxy

dAA

aa

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 15

Sample Problem 5.4• Using vertical strips, perform a single integration

to find the first moments.

1052

2

1

2

44

2

0

5

4

2

0

22

2

2

0

4

2

0

22

abx

a

b

dxxa

bdxy

ydAyQ

bax

a

b

dxxa

bxdxxydAxQ

a

a

elx

a

a

ely

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 16

Sample Problem 5.4• Or, using horizontal strips, perform a single

integration to find the first moments.

10

42

1

22

2

0

2321

2121

2

0

22

0

22

abdyy

b

aay

dyyb

aaydyxaydAyQ

badyy

b

aa

dyxa

dyxaxa

dAxQ

b

elx

b

b

ely

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 17

Sample Problem 5.4• Evaluate the centroid coordinates.

43

2baabx

QAx y

ax4

3

103

2ababy

QAy x

by10

3

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

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Distributed Loads on Beams

• A distributed load is represented by plotting the load per unit length, w (N/m) . The total load is equal to the area under the load curve. AdAdxwW

L

0

AxdAxAOP

dWxWOP

L

0

• A distributed load can be replaced by a concentrated load with a magnitude equal to the area under the load curve and a line of action passing through the centroid of the area.

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 19

Sample Problem 5.9

A beam supports a distributed load as shown. Determine the equivalent concentrated load and the reactions at the supports.

SOLUTION:

• The magnitude of the concentrated load is equal to the total load or the area under the curve.

• The line of action of the concentrated load passes through the centroid of the area under the curve.

• Determine the support reactions by summing moments about the beam ends.

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 20

Sample Problem 5.9SOLUTION:

• The magnitude of the concentrated load is equal to the total load or the area under the curve.

kN 0.18F

• The line of action of the concentrated load passes through the centroid of the area under the curve.

kN 18

mkN 63 X m5.3X

IZMIR INSTITUTE OF TECHNOLOGYIZMIR INSTITUTE OF TECHNOLOGY Department of ArchitectureDepartment of Architecture AR23AR2311 Fall12/13Fall12/13

11.04.23 Dr. Engin Aktaş 21

Sample Problem 5.9• Determine the support reactions by summing

moments about the beam ends.

0m .53kN 18m 6:0 yA BM

kN 5.10yB

0m .53m 6kN 18m 6:0 yB AM

kN 5.7yA