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1 n this appendix we discuss... 1 of 4 16-Sep-12 19:35 APPENDIX C In this appendix we discuss how to calculate the moment of inertia of an area. 1 The moment of inertia of an area was first introduced in a footnote in Chapter 8 as one of the family of integrals related to area. It was also mentioned in Chapter 10 as the measure of the cross-sectional area that relates the bending moment in a beam to the stress in the beam (the denominator in 10.8). Consider the area shown in Figure C.1. The moment of inertia of this area about the x axis is (C1a) and about the y axis is (C1b) The unit of the moment of inertia of area is the fourth power of length. Figure C.1 The integrals in C1a and C1b are also called the second area integrals. For reference, the first area integrals are used to find the geometric center or centroid of an area (x c, y c ), 2 and the (zero) area integral is used to find the area (A). 3

2 In this appendix we discuss... 2 of 4 16-Sep-12 19:35 Now consider that the moments of inertia have been calculated relative to axes located at the area's centroid; call these values I xc. The parallel axis theorem allows us to use these values of I xc to find the moments of inertia of the area relative to any other parallel axes by (C2a) (C2b) where x 1 and y 1 are the distances between axes, as depicted in Figure C.2. Figure C.2 A few notes about the moments of inertia of an area: If the area is simple (e.g., circular, rectangular), you can generally find the values of I xc in a reference table, such as Table A3.1. The tabulated data were found based on the application of C1a and C1b. Equations C2a and C2b can be used to find I x based on known values of I xc. Alternately, if I x are known, these equations can be rearranged to calculate I xc. The value of I x reflect the distribution of the area relative to coordinate axes in the plane of the area. This means that two areas may have the same cross-sectional area, but if their areas are distributed differently, they will have different moments of inertia. For example, the circular and square areas in Figure C.3 have been sized to have the same areas, but have different moments of inertia of their areas. Though not proven here, one finds that the sum (I x + I y ) is a constant, independent of the orientation of axes (Figure C.4). This sum is called the polar moment of inertia. Furthermore, there is one particular orientation of the axes in which the value of I x or I y is a maximum and the other is a minimum. The moment of inertia of an area that is composed of distinct parts of simple areas can be found by using the parallel axis theorem applied to each distinct part relative to the centroid for the composite area. Determining the centroid of a composite area was presented in Section 8.1 (equation 8.12). The ideas presented in this section on calculating the moment of inertia of area apply to both positive and negative areas (i.e., a hole). In the case of a negative area, a negative sign is used in front of the moment of inertia, as illustrated in Figure C.5. The moments of inertia of an area are related to another area used in calculations, the radius of gyration of an area. The radius of gyration of an area A about the x axis (r x ) and about the y axis (r y ) are defined to be

3 In this appendix we discuss... 3 of 4 16-Sep-12 19:35 (C3a) (C3b) Figure C.3 The circle and square have the same cross-sectional area, but different area moments of inertia. Figure C.4 For a given origin, the sum (I x + I y ) is a constant.

4 In this appendix we discuss... 4 of 4 16-Sep-12 19:35 Figure C.5 An example of calculating the moment of inertia of a composite area. Copyright 2007 John Wiley & Sons, Inc. All rights reserved.

5 Exercises 1 of 4 16-Sep-12 19:35 EXERCISES C1-C3. Use integration to evaluate the moments of inertia of area I x of the shaded area shown in EC1, EC2, EC3. EC1 EC2 EC3 C4-C6. Use integration to evaluate the moments of inertia I xc of the shaded area shown in EC1, EC2, EC3, respectively. C7-C10. Determine the moments of inertia of area I x of the shaded area shown in EC7, EC8, EC9, EC10 using the properties in Table A3.1.

6 Exercises 2 of 4 16-Sep-12 19:35 EC7 EC8 EC9 EC10 C11-C15. Determine the moments of inertia of area I x of the composite area shown in EC11, EC12, EC13, EC14, EC15.

7 Exercises 3 of 4 16-Sep-12 19:35 EC11 EC12 EC13 EC14 EC15

8 xercises 4 of 4 16-Sep-12 19:35 Copyright 2007 John Wiley & Sons, Inc. All rights reserved.

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