Ishik University / Sulaimani Civil Engineering Department. Engineering Mechanics CHAPTER -10- MOMENTS OF INERTIA. Chapter Objectives

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1 Ishik Universit / Sulaimani Civil Engineering Department Engineering Mechanics CHAPTER -10- MOMENTS OF INERTIA Copright 010 Pearson Education South Asia Pte Ltd 1 Chapter Objectives Method for determining the moment of inertia for an area. Introduce product of inertia and show determine the maimum and minimum moments of inertia for an area. Discuss the mass moment of inertia. Copright 010 Pearson Education South Asia Pte Ltd 1

2 Chapter Outline 1. Definitions of Moments of Inertia for Areas. Parallel-Ais Theorem for an Area 3. Moments of Inertia for Composite Areas Copright 010 Pearson Education South Asia Pte Ltd 3 1. Definition of Moments of Inertia for Areas Copright 010 Pearson Education South Asia Pte Ltd 4

1. Definition of Moments of Inertia for Areas Centroid for an area is determined b the first moment of an area about an ais Second moment of an area is referred as

Moment of inertia of an area originates whenever one relates the normal stress σ or force per unit area Copright 010 Pearson Education South Asia Pte Ltd 5 1.

3 1. Definition of Moments of Inertia for Areas Centroid for an area is determined b the first moment of an area about an ais Second moment of an area is referred as the moment of inertia. Moment of inertia of an area originates whenever one relates the normal stress σ or force per unit area Copright 010 Pearson Education South Asia Pte Ltd 5 1. Definition of Moments of Inertia for Areas Moment of Inertia Consider area A ling in the - plane B definition, moments of inertia of the differential plane area da about the and aes di For entire area, moments of inertia are given b I I da A A da da di da Copright 010 Pearson Education South Asia Pte Ltd 6 3

1. Definition of Moments of Inertia for Areas Moment of Inertia Formulate the second moment of da about the pole O or z ais This is known as the polar ais dj O r da where r is perpendicular from the

4 1. Definition of Moments of Inertia for Areas Moment of Inertia Formulate the second moment of da about the pole O or z ais This is known as the polar ais dj O r da where r is perpendicular from the pole (z ais) to the element da Polar moment of inertia for entire area, J O r da I A I Copright 010 Pearson Education South Asia Pte Ltd 7. Parallel Ais Theorem for an Area For moment of inertia of an area known about an ais passing through its centroid, determine the moment of inertia of area about a corresponding parallel ais using the parallel ais theorem. Consider moment of inertia of the shaded area. A differential element da is located at an arbitrar distance from the centroidal ais. Copright 010 Pearson Education South Asia Pte Ltd 8 4

. Parallel Ais Theorem for an Area The fied distance between the parallel and aes is defined as d For moment of inertia of da about ais di ' d da For entire area I A A ' ' d da d First integral

5 . Parallel Ais Theorem for an Area The fied distance between the parallel and aes is defined as d For moment of inertia of da about ais di ' d da For entire area I A A ' ' d da d First integral represent the moment of inertia of the area about the centroidal ais da A ' da d A da Copright 010 Pearson Education South Asia Pte Ltd 9. Parallel Ais Theorem for an Area Second integral = 0 since passes through the area s centroid C ' da da 0; 0 Third integral represents the total area A Similarl I I For polar moment of inertia about an ais perpendicular to the - plane and passing through pole O (z ais) J O I I J Ad Ad C Ad Copright 010 Pearson Education South Asia Pte Ltd 10 5

3. Moments of Inertia for Composite Areas Composite area consist of a series of connected simpler parts or shapes Moment of inertia of the composite area = algebraic sum of the

Parallel Ais Theorem Moment of inertia of each part is determined about its centroidal ais Copright 010 Pearson Education South Asia Pte Ltd 11 3.

6 3. Moments of Inertia for Composite Areas Composite area consist of a series of connected simpler parts or shapes Moment of inertia of the composite area = algebraic sum of the moments of inertia of all its parts Procedure for Analsis Composite Parts Divide area into its composite parts and indicate the centroid of each part to the reference ais Parallel Ais Theorem Moment of inertia of each part is determined about its centroidal ais Copright 010 Pearson Education South Asia Pte Ltd Moments of Inertia of Composite Areas The moment of inertia of a composite area A about a given ais is obtained b adding the moments of inertia of the component areas A 1, A, A 3,..., with respect to the same ais. 6

7 Eample -1- Copright 010 Pearson Education South Asia Pte Ltd 13 Copright 010 Pearson Education South Asia Pte Ltd 14 7

8 Eample -- Find the area moment of inertia for the rectangle about its centroidal aes and. Copright 010 Pearson Education South Asia Pte Ltd 15 Copright 010 Pearson Education South Asia Pte Ltd 16 8

9 Copright 010 Pearson Education South Asia Pte Ltd 17 Copright 010 Pearson Education South Asia Pte Ltd 18 9

10 Eample -3- Determine the moment of inertia of the area about the - ais. Copright 010 Pearson Education South Asia Pte Ltd 19 Copright 010 Pearson Education South Asia Pte Ltd 0 10

11 Eample -4- Determine the moment of inertia of the area about the - ais. Copright 010 Pearson Education South Asia Pte Ltd 1 Copright 010 Pearson Education South Asia Pte Ltd 11

12 Eample -5- Find the area moment of inertia for the circular area about its centroidal aes and, as well as its polar moment of inertia about its center. Copright 010 Pearson Education South Asia Pte Ltd 3 Copright 010 Pearson Education South Asia Pte Ltd 4 1

13 Problem -1- Determine the moment of inertia of the area about the - ais. Copright 010 Pearson Education South Asia Pte Ltd 5 Problem -- Determine the moment of inertia of the area about the - ais. Copright 010 Pearson Education South Asia Pte Ltd 6 13

14 Problem -3- Determine the moment of inertia of the area about the - ais. Copright 010 Pearson Education South Asia Pte Ltd 7 Problem -4- Determine the moment of inertia of the area about the - ais. Copright 010 Pearson Education South Asia Pte Ltd 8 14

15 Problem -5- Determine the moment of inertia of the area about the - ais. Copright 010 Pearson Education South Asia Pte Ltd 9 Problem -6- Determine the moment of inertia of the area about the - ais. Copright 010 Pearson Education South Asia Pte Ltd 30 15

16 ENGINEERING MECHANICS STATICS, R.C.HIBBELER, 1 TH EDITION, CHAPTER -10- FUNDAMENTAL PROBLEMS F10.1, F10., F10.3, F10.4 Copright 010 Pearson Education South Asia Pte Ltd 31 Eample -6- Compute the moment of inertia of the composite area about the ais. Copright 010 Pearson Education South Asia Pte Ltd 3 16

17 Composite Parts Composite area obtained b subtracting the circle form the rectangle. Centroid of each area is located in the figure below. Copright 010 Pearson Education South Asia Pte Ltd 33 Parallel Ais Theorem Circle I I 1 4 Rectangle I I 1 1 ' ' Ad mm Ad mm Copright 010 Pearson Education South Asia Pte Ltd 34 17

18 Summation For moment of inertia for the composite area, I mm Copright 010 Pearson Education South Asia Pte Ltd 35 18

13. Buckling of Columns *13.6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING

13. Buckling of Columns *13.6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING *13.6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING To account for behavior of different-length columns, design codes specify several formulae that will best fit the data within the short, intermediate, and