Jump to ContentJump to Main Navigation
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

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 28 February 2017

(p.317) Appendix D Parallel-Axis Theorem

(p.317) Appendix D Parallel-Axis Theorem

Engineering Mechanics of Deformable Solids
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 .

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

I y = I y y = A z 2 d A = I y c + A d z 2

I y z = I z y = A y z d A = I y c z c + A d y d z ,

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.