Slide 1ContentsIntroductionCenter of Gravity of a 2D BodyCentroids and First Moments of Areas and LinesFirst Moments of Areas and LinesCentroids of Common Shapes of AreasCentroids of Common Shapes of LinesComposite Plates and AreasSample Problem 5.1Slide 11Slide 12Determination of Centroids by IntegrationSample Problem 5.4Slide 15Slide 16Slide 17Slide 18Theorems of Pappus-GuldinusSlide 20Sample Problem 5.7Slide 22Distributed Loads on BeamsSample Problem 5.9Slide 25Slide 26Center of Gravity of a 3D Body: Centroid of a VolumeCentroids of Common 3D ShapesComposite 3D BodiesSample Problem 5.12Slide 31Slide 32VECTOR MECHANICS FOR ENGINEERS: STATICSEighth EditionFerdinand P. BeerE. Russell Johnston, Jr.Lecture Notes:J. Walt OlerTexas Tech UniversityCHAPTER© 2007 The McGraw-Hill Companies, Inc. All rights reserved. 5Distributed Forces: Centroids and Centers of Gravity© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 2ContentsIntroductionCenter of Gravity of a 2D BodyCentroids and First Moments of Areas and LinesCentroids of Common Shapes of AreasCentroids of Common Shapes of LinesComposite Plates and AreasSample Problem 5.1Determination of Centroids by IntegrationSample Problem 5.4Theorems of Pappus-GuldinusSample Problem 5.7Distributed Loads on BeamsSample Problem 5.9Center of Gravity of a 3D Body: Centroid of a VolumeCentroids of Common 3D ShapesComposite 3D BodiesSample Problem 5.12© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 3Introduction•The earth exerts a gravitational force on each of the particles forming a body. These forces can be replace by a single equivalent force equal to the weight of the body and applied at the center of gravity for the body.•The centroid of an area is analogous to the center of gravity of a body. The concept of the first moment of an area is used to locate the centroid.•Determination of the area of a surface of revolution and the volume of a body of revolution are accomplished with the Theorems of Pappus-Guldinus.© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 4Center of Gravity of a 2D Body•Center of gravity of a platedWyWyWyMdWxWxWxMyy•Center of gravity of a wire© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 5Centroids and First Moments of Areas and Lines   xQdAyAyyQdAxAxdAtxAtxdWxWxxy respect toh moment witfirst respect toh moment witfirst •Centroid of an area   dLyLydLxLxdLaxLaxdWxWx•Centroid of a line© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 6First Moments of Areas and Lines•An area is symmetric with respect to an axis BB’ if for every point P there exists a point P’ such that PP’ is perpendicular to BB’ and is divided into two equal parts by BB’.•The first moment of an area with respect to a line of symmetry is zero.•If an area possesses a line of symmetry, its centroid lies on that axis•If an area possesses two lines of symmetry, its centroid lies at their intersection.•An area is symmetric with respect to a center O if for every element dA at (x,y) there exists an area dA’ of equal area at (-x,-y). •The centroid of the area coincides with the center of symmetry.© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 7Centroids of Common Shapes of Areas© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 8Centroids of Common Shapes of Lines© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 9Composite Plates and Areas•Composite platesWyWYWxWX•Composite areaAyAYAxAX© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 10Sample Problem 5.1For the plane area shown, determine the first moments with respect to the x and y axes and the location of the centroid.SOLUTION:•Divide the area into a triangle, rectangle, and semicircle with a circular cutout.•Compute the coordinates of the area centroid by dividing the first moments by the total area.•Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout.•Calculate the first moments of each area with respect to the axes.© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 11Sample Problem 5.13333mm107.757mm102.506yxQQ•Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout.© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 12Sample Problem 5.12333mm1013.828mm107.757AAxXmm 8.54X2333mm1013.828mm102.506AAyYmm 6.36Y•Compute the coordinates of the area centroid by dividing the first moments by the total area.© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 13Determination of Centroids by Integration  ydxydAyAyydxxdAxAxelel2    dxxaydAyAydxxaxadAxAxelel2drrdAyAydrrdAxAxelel2221sin3221cos32dAydydxydAyAydAxdydxxdAxAxelel•Double integration to find the first moment may be avoided by defining dA as a thin rectangle or strip.© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: StaticsEighthEdition5 - 14Sample Problem 5.4Determine by direct integration the location of the centroid of a parabolic spandrel.SOLUTION:•Determine the constant k.•Evaluate the total area.•Using either vertical or horizontal strips, perform a single integration to find the first moments.•Evaluate the centroid coordinates.© 2007 The