Distributed Forces: Moments of nertia
Contents ntroduction Moments of nertia of an Area Moments of nertia of an Area b ntegration Polar Moments of nertia Radius of Gration of an Area Sample Problems Parallel Ais Theorem Moments of nertia of Composite Areas Sample Problems Product of nertia Principal Aes and Principal Moments of nertia Sample Problems Mohr s Circle for Moments and Products of nertia Sample Problem Moments of nertia of a Mass Parallel Ais Theorem Moments of nertia of Thin Plates Moments of nertia of a D Bod b ntegration Moments of nertia of Common Geometric Shapes Sample Problem Moments of nertia With Respect to an Arbitrar Ais Ellipsoid of nertia. Principle Aes of Aes of nertia of a Mass
ntroduction Previousl considered distributed forces which were proportional to the area or volume over which the act. - The resultant was obtained b summing or integrating over the areas or volumes. - The moments of the resultant about an ais was determined b computing the first moments of the areas or volumes about that ais. Will now consider forces which are proportional to the area or volume over which the act but also var linearl with distance from a given ais. - t will be shown that the magnitude of the resultant depends on the first moments of the force distribution with respect to the ais. - The point of application of the resultant depends on the second moments of the distribution with respect to the ais. Current chapter will present methods for computing the moments and products of inertia for areas and masses.
Moments of nertia of an Area Consider distributed forces F whose magnitudes are proportional to the elemental areas A on which the act and also var linearl with the distance of A from a given ais. Eample: Consider a beam subjected to pure bending. nternal forces var linearl with distance from the neutral ais which passes through the section centroid. F ka R M k k da da 0 da Q first moment da second moment Eample: Consider the net hdrostatic force on a submerged circular gate. F pa A R M da da
5 Moments of nertia of an Area b ntegration Second moments or moments of inertia of an area with respect to the and aes, da da Evaluation of the integrals is simplified b choosing da to be a thin strip parallel to one of the coordinate aes. For a rectangular area, da h 0 bd bh The formula for rectangular areas ma also be applied to strips parallel to the aes, d d d da d
6 Polar Moments of nertia The polar moments of inertia is an important parameter in problems involving torsion of clindrical shafts and rotations of slabs. J 0 r da The polar moments of inertia is related to the rectangular moments of inertia, J r da da da 0 da
Radius of Gration of an Area Consider area A with moments of inertia. magine that the area is concentrated in a thin strip parallel to the ais with equivalent. A k A k k = radius of gration with respect to the ais Similarl, A J k A k J A k A k O O O O O k k k 7
8 Sample Problem SOLUTON: A differential strip parallel to the ais is chosen for da. d da da l d Determine the moments of inertia of a triangle with respect to its base. For similar triangles, l b h h h l b h ntegrating d from = 0 to = h, da b h h h 0 h b d h h 0 h da b d h b h h 0 h d bh
9 Sample Problem SOLUTON: An annular differential area element is chosen, dj J O O u dj da O r 0 u da u du u du J O r 0 u r du a) Determine the centroidal polar moments of inertia of a circular area b direct integration. b) Using the result of part a, determine the moments of inertia of a circular area with respect to a diameter. From smmetr, =, JO r diameter r
0 Parallel Ais Theorem Consider moments of inertia of an area A with respect to the ais AA da The ais BB passes through the area centroid and is called a centroidal ais. da da d d da da d da Ad parallel ais theorem
Parallel Ais Theorem Moments of inertia T of a circular area with respect to a tangent to the circle, Ad r r r T 5 r Moments of inertia of a triangle with respect to a centroidal ais, AA BB 6 BB AA bh Ad Ad bh bh h
Moments of nertia of Composite Areas The moments of inertia of a composite area A about a given ais is obtained b adding the moments of inertia of the component areas A, A, A,..., with respect to the same ais.
Moments of nertia of Composite Areas
Sample Problem The strength of a W8 rolled steel beam is increased b attaching a plate to its upper flange. Determine the moments of inertia and radius of gration with respect to an ais which is parallel to the plate and passes through the centroid of the section. SOLUTON: Determine location of the centroid of composite section with respect to a coordinate sstem with origin at the centroid of the beam section. Appl the parallel ais theorem to determine moments of inertia of beam section and plate with respect to composite section centroidal ais. Calculate the radius of gration from the moments of inertia of the composite section.
5 SOLUTON: Determine location of the centroid of composite section with respect to a coordinate sstem with origin at the centroid of the beam section. Section Plate Beam Section A, in 6.75.0 A 7.95, in. 7.5 0 A, in 50. 0 A 50. Y A A Y A A 50. in 7.95 in.79 in.
Appl the parallel ais theorem to determine moments of inertia of beam section and plate with respect to composite section centroidal ais.,beam section,plate 7. in AY Ad 5. in 85.0.79 9 6.757.5.79, beam section,plate 7. 5. 68 in Calculate the radius of gration from the moments of inertia of the composite section. k A 67.5 in 7.95 in k 5.87 in. 6
7 Sample Problem SOLUTON: Compute the moments of inertia of the bounding rectangle and half-circle with respect to the ais. Determine the moments of inertia of the shaded area with respect to the ais. The moments of inertia of the shaded area is obtained b subtracting the moments of inertia of the half-circle from the moments of inertia of the rectangle.
8 SOLUTON: Compute the moments of inertia of the bounding rectangle and half-circle with respect to the ais. Rectangle: 6 00 8. 0 bh mm r 90 a 8. mm b 0 - a 8.8 mm A r.70 90 mm Half-circle: Moments of inertia with respect to AA, 6 r 90 5.760 mm AA 8 8 Moments of inertia with respect to, AA Aa 7.00 6 mm 6 5.760.70 Moments of inertia with respect to, Ab 9.0 6 mm 7.00 6.70 8.8
9 The moments of inertia of the shaded area is obtained b subtracting the moments of inertia of the half-circle from the moments of inertia of the rectangle. 8. 0 6 mm 9.0 6 mm 5.90 6 mm
0 Product of nertia Product of nertia: da When the ais, the ais, or both are an ais of smmetr, the product of inertia is zero. Parallel ais theorem for products of inertia: A
Principal Aes and Principal Moments of nertia The change of aes ields cos cos sin cos sin sin Given da da We wish to determine moments and product of inertia with respect to new aes and. Note: cos sin cos sin da The equations for and are the parametric equations for a circle, ave ave R R The equations for and lead to the same circle.
Principal Aes and Principal Moments of nertia At the points A and B, = 0 and is a maimum and minimum, respectivel. ma, min tan m ave The equation for Q m defines two angles, 90 o apart which correspond to the principal aes of the area about O. R ave ave R R ma and min are the principal moments of inertia of the area about O.
Sample Problem SOLUTON: Determine the product of inertia using direct integration with the parallel ais theorem on vertical differential area strips Appl the parallel ais theorem to evaluate the product of inertia with respect to the centroidal aes. Determine the product of inertia of the right triangle (a) with respect to the and aes and (b) with respect to centroidal aes parallel to the and aes.
SOLUTON: Determine the product of inertia using direct integration with the parallel ais theorem on vertical differential area strips b h d b h d da b h el el ntegrating d from = 0 to = b, b b b el el b b h d b b h d b h da d 0 0 0 8 h b
5 Appl the parallel ais theorem to evaluate the product of inertia with respect to the centroidal aes. b With the results from part a, b h h A b h bh b h 7
6 Sample Problem SOLUTON: Compute the product of inertia with respect to the aes b dividing the section into three rectangles and appling the parallel ais theorem to each. Determine the orientation of the principal aes (Eq. 9.5) and the principal moments of inertia (Eq. 9. 7). For the section shown, the moments of inertia with respect to the and aes are = 0.8 in and = 6.97 in. Determine (a) the orientation of the principal aes of the section about O, and (b) the values of the principal moments of inertia about O.
7 SOLUTON: Compute the product of inertia with respect to the aes b dividing the section into three rectangles. Appl the parallel ais theorem to each rectangle, A Note that the product of inertia with respect to centroidal aes parallel to the aes is zero for each rectangle. Rectangle Area, in.5.5.5, in..5 0.5, in..75 0.75 A A,in.8 0.8 6.56 A 6.56 in
8 Determine the orientation of the principal aes (Eq. 9.5) and the principal moments of inertia (Eq. 9. 7). tan m m 75. and 55. 6.56 0.8 6.97.85 m 7.7 and m 7. 7 0.8 in 6.97 in 6.56 in ma,min 0.8 6.97 0.8 6.97 6.56 a b ma min 5.5 in.897 in
9 Mohr s Circle for Moments and Products of nertia The moments and product of inertia for an area are plotted as shown and used to construct Mohr s circle, ave R Mohr s circle ma be used to graphicall or analticall determine the moments and product of inertia for an other rectangular aes including the principal aes and principal moments and products of inertia.
0 Sample Problem The moments and product of inertia with respect to the and aes are = 7.06 mm, =.606 mm, and = -.50 6 mm. Using Mohr s circle, determine (a) the principal aes about O, (b) the values of the principal moments about O, and (c) the values of the moments and product of inertia about the and aes SOLUTON: Plot the points (, ) and (,- ). Construct Mohr s circle based on the circle diameter between the points. Based on the circle, determine the orientation of the principal aes and the principal moments of inertia. Based on the circle, evaluate the moments and product of inertia with respect to the aes.
SOLUTON: Plot the points (, ) and (,- ). Construct Mohr s circle based on the circle diameter between the points. OC CD R ave.50.950 6 CD DX.7 0 mm 6 mm 6 mm 7.0.60 6 6.5 0 mm mm 6 mm Based on the circle, determine the orientation of the principal aes and the principal moments of inertia. tan m DX.097 m 7. 6 CD m. 8 ma OA ave R ma 8.60 6 mm min OB ave R min.90 6 mm
Based on the circle, evaluate the moments and product of inertia with respect to the aes. The points X and Y corresponding to the and aes are obtained b rotating CX and CY counterclockwise through an angle Q (60 o ) = 0 o. The angle that CX forms with the aes is f = 0 o - 7.6 o = 7. o. OF OC CX cos Rcos7. ' OG OC CYcos Rcos7. ' ave 5.960 ave 6 mm o o.890 6 mm FX CYsin Rsin 7. ' o OC R ave.7 0.950 6 mm 6 mm.80 6 mm
Moments of nertia of a Mass Angular acceleration about the ais AA of the small mass m due to the application of a couple is proportional to r m. r m = moments of inertia of the mass m with respect to the ais AA For a bod of mass m the resistance to rotation about the ais AA is r m r m r m r dm mass moment of inertia The radius of gration for a concentrated mass with equivalent mass moments of inertia is k m k m
Moments of nertia of a Mass Moments of inertia with respect to the coordinate ais is r dm z dm Similarl, for the moments of inertia with respect to the and z aes, z z n S units, r dm dm dm kg m n U.S. customar units, lb s slug ft ft ft lb ft s
Parallel Ais Theorem For the rectangular aes with origin at O and parallel centroidal aes, dm z dm z z dm dm z dm z z dm z z m m m z z z Generalizing for an ais AA and a parallel centroidal ais, md 5
6 Moments of nertia of Thin Plates For a thin plate of uniform thickness t and homogeneous material of densit, the mass moments of inertia with respect to ais AA contained in the plate is AA r t dm t AA, area r da Similarl, for perpendicular ais BB which is also contained in the plate, BB t BB, area For the ais CC which is perpendicular to the plate, CC AA BB t JC, area t AA, area BB, area
7 Moments of nertia of Thin Plates For the principal centroidal aes on a rectangular plate, AA BB CC t t AA BB a b, area t ma ab, area t mb AA, mass BB, mass m a b For centroidal aes on a circular plate, AA BB t AA r, area t mr CC AA BB mr
8 Moments of nertia of a D Bod b ntegration Moments of inertia of a homogeneous bod is obtained from double or triple integrations of the form r dv For bodies with two planes of smmetr, the moments of inertia ma be obtained from a single integration b choosing thin slabs perpendicular to the planes of smmetr for dm. The moments of inertia with respect to a particular ais for a composite bod ma be obtained b adding the moments of inertia with respect to the same ais of the components.
Moments of nertia of Common Geometric Shapes 9
0 Sample Problem SOLUTON: With the forging divided into a prism and two clinders, compute the mass and moments of inertia of each component with respect to the z aes using the parallel ais theorem. Add the moments of inertia from the components to determine the total moments of inertia for the forging. Determine the moments of inertia of the steel forging with respect to the z coordinate aes, knowing that the specific weight of steel is 90 lb/ft.
SOLUTON: Compute the moments of inertia of each component with respect to the z aes. clindersa in., L in.,.5in., in. : ma m 0.089 0.089.590 lb ft s each clinder : V m g m 0.089 lb s 90lb/ft in 78in ft.ft s ft m a 0.089 0.089.5.7 0 m L m lb ft s a L m 0.089 0.089.5 6.80 lb ft s
prism : m V g m 0. lb s 90lb/ft 6 78in ft.ft s ft in prism (a = in., b = 6 in., c = in.): z.880 c 0. 6 m b lb ft s a 0. m c 0.977 0 lb ft s Add the moments of inertia from the components to determine the total moments of inertia. z.880 0.977 0.880.590.7 0 6.80 z 0.060 lb ft s 9.0 lb ft s 7.80 lb ft s
Moments of nertia With Respect to an Arbitrar Ais OL = moments of inertia with respect to ais OL OL p dm r dm Epressing r and in terms of the vector components and epanding ields OL z z z The definition of the mass products of inertia of a mass is an etension of the definition of product of inertia of an area dm m z z z dm z dm z z mz mz z z z
Ellipsoid of nertia. Principal Aes of nertia of a Mass Assume the moments of inertia of a bod has been computed for a large number of aes OL and that point Q is plotted on each ais at a distance OQ The locus of points Q forms a surface known as the ellipsoid of inertia which defines the moments of inertia of the bod for an ais through O.,,z aes ma be chosen which are the principal aes of inertia for which the products of inertia are zero and the moments of inertia are the principal moments of inertia. OL