Radius of Gyration of an Area

Engineering Mechanics: Statics Fourteenth Edition Chapter 10 Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Definition of Moments of Inertia for Areas, Radius of Gyration of an Area Today’s Objectives: a. Define the moments of inertia (MoI) for an area. b. Determine the MoI for an area by integration. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Section 10.1: Definition of Moments of Inertia for Areas For the given vertical loading P on the beam shown on the right, which shape will develop less internal stress and deflection? Why? The answer depends on the MoI of the beam about the x-axis. It turns out that Section A has the highest MoI because most of the area is farthest from the x axis. Hence, it has the least stress and deflection. For the differential area dA shown in the figure: d Ix = y 2 dA, d Iy = x 2 dA, and, d JO = r 2 dA, where JO is the polar moment of inertia about the pole O or z axis. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Section 10.1: Definition of Moments of Inertia for Areas The moments of inertia for the entire area are obtained by integration. Ix =  y 2 dA ; Iy = A  A x 2 dA JO =  r 2 dA =  (x 2 + y 2 )dA = Ix + Iy A A The MoI is also referred to as the second moment of an area and has units of length to the fourth power (m4 or in4 ). Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved MoI for an Area by Integration (1 of 3) For simplicity, the area element used has a differential size in only one direction (dx or dy). This results in a single Integration and is usually simpler than doing a double integration with two differentials, i.e., dx  dy. The step-by-step procedure is: 1. Choose the element dA : There are two choices: a vertical strip or a horizontal strip. Some considerations about this choice are: Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved MoI for an Area by Integration (2 of 3) a. The element parallel to the axis about which the MoI is to be determined usually results in an easier solution. For example, we typically choose a horizontal strip for determining Ix and a vertical strip for determining Iy . b. If y is easily expressed in terms of x (e.g., y = x 2 + 1), then choosing a vertical strip with a differential element dx wide may be advantageous. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved MoI for an Area by Integration (3 of 3) 2. Integrate to find the MoI. For example, given the element shown in the figure above: Iy =  x 2 dA =  x 2 y dx  1 Ix =  dIx =    y 3 dx 3 (using the parallel axis theorem as per Example 10.2 of the textbook). Since the differential element is dx, y needs to be expressed in terms of x and the integral limit must also be in terms of x. As you can see, choosing the element and integrating can be challenging. It may require a trial and error approach, plus experience. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Example (1 of 2) Given: The shaded area shown in the figure. Find: The MoI of the area about the x- and y-axes. Plan: Follow the steps given earlier. Solution: Ix = y 2 dA 3 2 dA = (1 − x)dy = (1 − y )dy 1 Ix = y 2 3 2 (1 − y )dy 0 1  1  3  2  92  =   y −   y  = 0.111m4  9  0  3  Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Example (2 of 2) Iy = x = = 2 dA = x 2 2 3 1 8 3 x x 2 y dx (x ) dx dx 0 1  3  11  3 =  x  11   0 = 0.273 m 4 In the above example, Ix can be also determined using a vertical strip.  1 3 I = Then x    y dx = 3 1 1  1 2 4 x dx = = 0.111 m 0  3  9 Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Concept Quiz (2 of 2) 2. In the figure to the right, what is the differential moment of inertia of the element with respect to the y-axis A. x 2 y dx B.  1 3   x dy  12  C. y 2 x dy D.  1   y dy 3 (dIy )? Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Determine the moment of inertia for the shaded area about the x axis. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Determine the moment of inertia for the shaded area about the x axis. Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Attention Quiz (1 of 2) 1. When determining the MoI of the element in the figure, dIy equals A. x 2 dy B. x 2 dx C.  1 3   y dx 3 D. x 2.5 dx Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Attention Quiz (2 of 2) 2. Similarly, dIx equals A.  1  1.5   x dx 3 B. y 2 dA C.  1 3   x dy  12  D.  1 3   x dx 3 Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Reading Quiz (1 of 2) 1. The definition of the Moment of Inertia for an area involves an integral of the form A. B. C. D.  x dA.  x dA.  x dm.  m dA. 2 2 Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved Reading Quiz (2 of 2) 2. Select the correct SI units for the Moment of Inertia for an area. A. m3 B. m4 C. kg  m2 D. kg  m3 Copyright © 2016, 2013, 2010 Pearson Education, Inc. All Rights Reserved