Area Moment of Inertia Composite Areas - joinville.ifsc.edu.brmigueltbahia/Estatica_2018/36 Area...

Engineering Mechanics: Statics * Dr. Yiheng Wang

Area Moment of InertiaComposite Areas

Objective of this video:

To demonstrate through example the step-by-step procedure to determine the area moment of inertia for composite areas.



Area Moment of InertiaComposite Areas

x

①②

③

321 AAAA

xxxx IIII ,3,2,1


http://en.wikipedia.org/wiki/List_of_area_moments_of_inertia 04/14/2013

List of Area Moments of Inertia


http://en.wikipedia.org/wiki/List_of_area_moments_of_inertia

200 mm

100 mm

260 mm

20 mm 20 mm

20 mm

20 mm

xC

Example 1: Find the area moment of inertia of the composite area about its centroidal axis, x.


200 mm

100 mm

260 mm

20 mm 20 mm

20 mm

Step 1: Define the component areas.

① ②③

④

20 mm


200 mm

100 mm

260 mm

20 mm 20 mm

20 mm

Step 2: Choose a reference line.

reference line

y = 0

① ②③

④

20 mm


200 mm

100 mm

260 mm

20 mm 20 mm

20 mm

reference line

y = 0

① ②③

④

20 mm

Step 3: Determine the centroid

location of the composite area:

A

Ayy

~

A1 = A2 = 2000 mm2

A3 = 4000 mm2

A4 = 5200 mm2

∑A = 13200 mm2


reference line

y = 0



A

Ayy

~

① ②③

④

mm230~~21 yy mm270~

3 y

mm130~4 y

A1 = A2 = 2000 mm2

A3 = 4000 mm2

A4 = 5200 mm2

∑A = 13200 mm2

mm7.202


reference line

y = 0



A

Ayy

~

① ②③

④

mm7.202y

A1 = A2 = 2000 mm2

A3 = 4000 mm2

A4 = 5200 mm2xC

∑A = 13200 mm2


reference line

y = 0

① ②③

④

mm7.202y

A1 = A2 = 2000 mm2

A3 = 4000 mm2

A4 = 5200 mm2xC

Step 4: Determine the area moment of inertia

of each component area about x axis using

parallel axis theorem.

2

' AdII xx

∑A = 13200 mm2





2

' AdII xx

① ②③

④

mm7.202y

A1 = A2 = 2000 mm2xC

mm230~~21 yy

100 mm

20 mm

①

4633

',1 mm10667.11002012

1

12

1 bhI x

mm3.277.2022301 d

4626

,2,1 mm10158.33.27200010667.1 xx II

x’





2

' AdII xx

① ②③

④

mm7.202y

A3 = 4000 mm2xC

4633

',3 mm101333.02020012

1

12

1 bhI x

mm3.677.2022703 d

4626

,3 mm1025.183.674000101333.0 xI

20 mm ③

200 mm

mm270~3 y

x’





2

' AdII xx

① ②③

④

mm7.202y

A4 = 5200 mm2xC

4633

',4 mm1029.292602012

1

12

1 bhI x

mm7.721307.2024 d

4626

,4 mm1077.567.7252001029.29 xI

260 mm

20 mm

mm130~4 y

④

x’


Step 5: Add up the area moment of inertia of component about x axis.

4646 m1034.81mm1034.81 xI

① ②③

④

xC46

,4 mm1077.56 xI

46

,3 mm1025.18 xI

46

,2,1 mm10158.3 xx II


200 mm

100 mm

260 mm

20 mm 20 mm

20 mm

20 mm

x’’

C

Example 2: Find the area moment of inertia of the composite area about the x” axis.


46 mm1034.81 xI

xC

x”

mm7.202y

46

26

"

mm107.236

7.202132001034.81

xI

∑A = 13200 mm2

d = 202.7 mm


Question 1: Determine the area moment of inertia of the shaded area about the x axis.

x

y

4 ft

3 ft

3 ft

1 ft

2 ft4 ft

(a) 149 ft4

(b) 221 ft4

(c) 184 ft4

(d) 117 ft4


Question 2: Determine the area moment of inertia of the shaded area about the y axis.

(a) 149 ft4

(b) 221 ft4

(c) 184 ft4

(d) 117 ft4

x

y

4 ft

3 ft

3 ft

1 ft

2 ft4 ft


Area Moment of Inertia Composite Areas - joinville.ifsc.edu.brmigueltbahia/Estatica_2018/36 Area...

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