Ninth Edition CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinand P. Beer E. Russell Johnston, Jr. 9 Distributed Forces: Lecture Notes: Moments of Inertia J. Walt Oler Texas Tech University © 2010 The McGraw-Hill Companies, Inc. All rights reserved. Edition Ninth Vector Mechanics for Engineers: Statics Contents Introduction Sample Problem 9.6 Moments of Inertia of an Area Sample Problem 9.7 Moment of Inertia of an Area by Mohr’s Circle for Moments and Integration Products of Inertia Polar Moment of Inertia Sample Problem 9.8 Radius of Gyration of an Area Moment of Inertia of a Mass Sample Problem 9.1 Parallel Axis Theorem Sample Problem 9.2 Moment of Inertia of Thin Plates Parallel Axis Theorem Moment of Inertia of a 3D Body by Moments of Inertia of Composite Integration Areas Moment of Inertia of Common Sample Problem 9.4 Geometric Shapes Sample Problem 9.5 Sample Problem 9.12 Product of Inertia Moment of Inertia With Respect to an Principal Axes and Principal Arbitrary Axis Moments of Inertia Ellipsoid of Inertia. Principle Axes of Axes of Inertia of a Mass © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 2 Edition Ninth Vector Mechanics for Engineers: Statics Introduction • Previously considered distributed forces which were proportional to the area or volume over which they act. - The resultant was obtained by summing or integrating over the areas or volumes. - The moment of the resultant about any axis was determined by computing the first moments of the areas or volumes about that axis. • Will now consider forces which are proportional to the area or volume over which they act but also vary linearly with distance from a given axis. - It will be shown that the magnitude of the resultant depends on the first moment of the force distribution with respect to the axis. - The point of application of the resultant depends on the second moment of the distribution with respect to the axis. • Current chapter will present methods for computing the moments and products of inertia for areas and masses. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 3 Edition Ninth Vector Mechanics for Engineers: Statics Moment of Inertia of an Area • Consider distributed forces F whose magnitudes are proportional to the elemental areas A on which they act and also vary linearly with the distance of from a given axis. • Example: Consider a beam subjected to pure bending. Internal forces vary linearly with distance from the neutral axis which passes through the section centroid. F kyA R k y dA 0 y dA Qx first moment M k y2dA y2dA second moment • Example: Consider the net hydrostatic force on a submerged circular gate. F pA yA R y dA 2 M x y dA © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 4 Edition Ninth Vector Mechanics for Engineers: Statics Moment of Inertia of an Area by Integration • Second moments or moments of inertia of an area with respect to the x and y axes, 2 2 I x y dA I y x dA • Evaluation of the integrals is simplified by choosing dA to be a thin strip parallel to one of the coordinate axes. • For a rectangular area, h I y2dA y2bdy 1 bh3 x 3 0 • The formula for rectangular areas may also be applied to strips parallel to the axes, dI 1 y3dx dI x2dA x2 y dx x 3 y © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 5 Edition Ninth Vector Mechanics for Engineers: Statics Polar Moment of Inertia • The polar moment of inertia is an important parameter in problems involving torsion of cylindrical shafts and rotations of slabs. 2 J0 r dA • The polar moment of inertia is related to the rectangular moments of inertia, 2 2 2 2 2 J0 r dA x y dA x dA y dA I y I x © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 6 Edition Ninth Vector Mechanics for Engineers: Statics Radius of Gyration of an Area • Consider area A with moment of inertia Ix. Imagine that the area is concentrated in a thin strip parallel to the x axis with equivalent Ix. I I k 2 A k x x x x A kx = radius of gyration with respect to the x axis • Similarly, I I k 2 A k y y y y A J J k 2 A k O O O O A 2 2 2 kO kx k y © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 7 Edition Ninth Vector Mechanics for Engineers: Statics Sample Problem 9.1 SOLUTION: • A differential strip parallel to the x axis is chosen for dA. 2 dIx y dA dA l dy • For similar triangles, l h y h y h y l b dA b dy b h h h Determine the moment of inertia of a triangle with respect • Integrating dI from y = 0 to y = h, to its base. x h h 2 2 h y b 2 3 Ix y dA y b dy hy y dy 0 h h 0 3 4 h b y y bh3 h h 3 4 I x 0 12 © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 8 Edition Ninth Vector Mechanics for Engineers: Statics Sample Problem 9.2 SOLUTION: • An annular differential area element is chosen, 2 dJO u dA dA 2 u du r r 2 3 JO dJO u 2 u du 2 u du 0 0 J r4 O 2 a) Determine the centroidal polar • From symmetry, I = I , moment of inertia of a circular x y area by direct integration. J I I 2I r4 2I O x y x 2 x b) Using the result of part a, determine the moment of inertia I I r4 of a circular area with respect to a diameter x 4 diameter. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 9 Edition Ninth Vector Mechanics for Engineers: Statics Parallel Axis Theorem • Consider moment of inertia I of an area A with respect to the axis AA’ I y2dA • The axis BB’ passes through the area centroid and is called a centroidal axis. I y2dA y d 2 dA y2dA 2d ydA d 2 dA I I Ad 2 parallel axis theorem © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 10 Edition Ninth Vector Mechanics for Engineers: Statics Parallel Axis Theorem • Moment of inertia IT of a circular area with respect to a tangent to the circle, I I Ad 2 1 r 4 r 2 r 2 T 4 5 r 4 4 • Moment of inertia of a triangle with respect to a centroidal axis, 2 I AA I BB Ad 2 I I Ad 2 1 bh3 1 bh 1 h BB AA 12 2 3 1 bh3 36 © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 11 Edition Ninth Vector Mechanics for Engineers: Statics Moments of Inertia of Composite Areas • The moment of inertia of a composite area A about a given axis is obtained by adding the moments of inertia of the component areas A1, A2, A3, ... , with respect to the same axis. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 12 Edition Ninth Vector Mechanics for Engineers: Statics Moments of Inertia of Composite Areas © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 13 Edition Ninth Vector Mechanics for Engineers: Statics Sample Problem 9.4 SOLUTION: • Determine location of the centroid of composite section with respect to a coordinate system with origin at the centroid of the beam section. • Apply the parallel axis theorem to determine moments of inertia of beam section and plate with respect to The strength of a W14x38 rolled steel composite section centroidal axis. beam is increased by attaching a plate • Calculate the radius of gyration from the to its upper flange. moment of inertia of the composite Determine the moment of inertia and section. radius of gyration with respect to an axis which is parallel to the plate and passes through the centroid of the section. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 14 Edition Ninth Vector Mechanics for Engineers: Statics Sample Problem 9.4 SOLUTION: • Determine location of the centroid of composite section with respect to a coordinate system with origin at the centroid of the beam section. Section A, in 2 y, in. yA, in3 Plate 6.75 7.425 50.12 Beam Section 11.20 0 0 A 17.95 yA 50.12 yA 50.12 in3 Y A yA Y 2.792 in. A 17.95 in 2 © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 15 Edition Ninth Vector Mechanics for Engineers: Statics Sample Problem 9.4 • Apply the parallel axis theorem to determine moments of inertia of beam section and plate with respect to composite section centroidal axis. 2 2 Ix,beam section Ix AY 385 11.202.792 472.3 in 4 3 I I Ad 2 1 9 3 6.75 7.425 2.792 2 x,plate x 12 4 145.2 in 4 Ix Ix,beam section Ix,plate 472.3 145.2 4 Ix 618 in • Calculate the radius of gyration from the moment of inertia of the composite section. 4 Ix 617.5 in kx kx 5.87 in. A 17.95 in 2 © 2010 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 16 Edition Ninth Vector Mechanics for Engineers: Statics Sample Problem 9.5 SOLUTION: • Compute the moments of inertia of the bounding rectangle and half-circle with respect to the x axis.