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Engineering Mechanics of Deformable SolidsA Presentation with Exercises$

Sanjay Govindjee

Print publication date: 2012

Print ISBN-13: 9780199651641

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199651641.001.0001

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(p.317) Appendix D Parallel-Axis Theorem

(p.317) Appendix D Parallel-Axis Theorem

Source:
Engineering Mechanics of Deformable Solids
Publisher:
Oxford University Press

The area moments of inertia of a given area A are defined as

I z = I z z = A y 2 d A , I y = I y y = A z 2 d A , I z y = I y z = A y z d A .
(D.1)

If the area moments of inertia (Izc , Iyc , Iyc zc ) of an area are known with respect to the centroidal axes of the area, then the parallel-axis theorem tells us that the area moments of inertia with respect to any other set of (parallel) axes are given by:

I z = I z z = A y 2 d A = I z c + A d y 2
(D.2)

I y = I y y = A z 2 d A = I y c + A d z 2
(D.3)

I y z = I z y = A y z d A = I y c z c + A d y d z ,
(D.4)

where dy and dz are the (centroidal) coordinates of the y-z frame (see Fig. D.1). The proof of the theorem follows directly from eqn (D.1) under the substitution y = yc dy and z = zc dz , and the fact that ∫ A zc dA = ∫ A yc dA = 0 by definition of the centroid.


                     Appendix D
                   Parallel-Axis Theorem

Fig. D.1 Parallel-axis theorem construction. The axes yc and zc represent the centroidal axes.