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MSU ME 221 - Lecture 18 Statics

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PowerPoint PresentationHomework #7Exam #2 ResultsSecond Moments of AreaMoments of AreaMoments of Basic ShapesPolar MomentParallel Axis TheoremRadius of GyrationPrincipal Second MomentsProduct Moment of InertiaMohr’s Circle for Principal InertiaExampleME 221 Lecture 18 1ME 221 StaticsLecture #18Sections 9.1 – 9.6ME 221 Lecture 18 2Homework #7•Chapter 4 problems:–52, 53 & 54•Chapter 9 problems:–2, 3, 4, 11, 23, 29, 32–Use double integration for 2, 3, 4 & 11•Due Friday, October 17ME 221 Lecture 18 3Exam #2 ResultsAverage = xx.xScores posted on AngelSolution will be posted on Angel todaySee syllabus for regrade policyLast day to drop without a grade reported is: Tuesday, October 14ME 221 Lecture 18 4Second Moments of Area•Second moments of area play a central role in mechanics of materials and dynamics•Definition of second moment–Basic areas (rectangle, circular, triangular)•Definition of polar moment–Basic areas (circular)•Parallel axis theoremME 221 Lecture 18 5Moments of Area•Normally, we want the moment with respect to centriod axes•Moment of inertia–Moment about other axes derived from centroid casedAyxr2 2 and xx yyA AI y dA I x dA  ME 221 Lecture 18 6Moments of Basic Shapes•Rectanglexy2 22 22b hb hxxI y dy dx     223 32 2=3 3bbh hdx    3=12bh•Circularxy 220 0sinRxx yyI I r rdr d   24 2140sinR d 414RdA = r dr dME 221 Lecture 18 7Polar MomentThe polar moment is the second moment of area about the z-axisxyr23 4120 0ROzJ r dr d R   Note that: Ixx + Iyy = JOzyyxxozIIdAyxdArJ  )(222ME 221 Lecture 18 8Parallel Axis TheoremThe centroid of the area MUSTMUST be one of the axes used in the parallel axis theorem.ydyxx’C22xx xxc yyy yyc xI I AdI I Ad  ME 221 Lecture 18 9Radius of GyrationAn alternate, equivalent way to represent the moment of an area; ;yyxx Ozx y zII Jk k kA A A  Distance from the point or axis to where the area is concentratedME 221 Lecture 18 10Principal Second Moments•Definition of product moment of inertia•Definition of principal axes–Product of inertia axis theorem•Mohr’s circle to find principal axes•ExampleME 221 Lecture 18 11Product Moment of Inertia•Basic section with two axes of symmetryxy•Composite sections - product parallel axis theoremxyAI xydAxy x y o oI I x y A  ME 221 Lecture 18 12Mohr’s Circle for Principal Inertia+Ixy-IxyIxx, IyyIyyxIxxIxyx• draw point (Ixx , Ixy) and the circle(Ixx+Iyy)/2• draw circle center (Ixx + Iyy)/2xIMAXIMIN2• use geometry to find IMAX , IMIN and ANGLES IN MOHR’S CIRCLE ARE TWICE THOSE IN THE CROSS SECTION!!!ME 221 Lecture 18


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