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ME 221 Statics Lecture #9 Sections 9.1 – 9.6

ME 221 Statics Lecture #9 Sections 9.1 – 9.6. Homework #4. Chapter 4 problems: 52 & 54 Chapter 9 problems 2, 11 & 32 Due Monday, June 14 MatLab Group Problems 4.13 – plot as a function of R with 0 º < α < 45º 4.51 – plot as a function of length “ l ”

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ME 221 Statics Lecture #9 Sections 9.1 – 9.6

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  1. ME 221 Statics Lecture #9Sections 9.1 – 9.6 Lecture 9

  2. Homework #4 • Chapter 4 problems: • 52 & 54 • Chapter 9 problems • 2, 11 & 32 • Due Monday, June 14 • MatLab Group Problems • 4.13 – plot as a function of R with 0º < α < 45º • 4.51 – plot as a function of length “l” • 4.60 – plot the equivalent force and point of action • Due Monday, June 14 Lecture 9

  3. Exam #1 Results Scores posted on Angel Solution posted on Angel See syllabus for regrade policy Last day to drop without a grade reported is: Wednesday, June 9 Lecture 9

  4. Second Moments of Area(Area Moments of Inertia) • 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 theorem Lecture 9

  5. x dA y r Moments of Area • Characterize distribution of area about the centroid • Normally, we want the moment with respect to centriod axes • Moment about other axes derived from centroid case • Moment of inertia Lecture 9

  6. y x dA = r dr dq y x Moments of Basic Shapes • Rectangle • Circular Lecture 9

  7. y r x Polar Moment The polar moment is the second moment of area about the z-axis Note that: Ixx + Iyy = JOz Lecture 9

  8. y x’ C dy x Parallel Axis Theorem The centroid of the area MUST be one of the axes used in the parallel axis theorem. Lecture 9

  9. Radius of Gyration An alternate, equivalent way to represent the moment of an area Distance from the point or axis to where the area is concentrated Lecture 9

  10. Principal Second Moments • Definition of product moment of inertia • Product of inertia axis theorem • Definition of principal axes • Mohr’s circle to find principal axes • Example Lecture 9

  11. y x Product Moment of Inertia(Measures Antisymmetry) • Basic section with two axes of symmetry • Composite sections - product parallel axis theorem Lecture 9

  12. -Ixy • draw circle center (Ixx + Iyy)/2 x • draw point (Ixx , Ixy) and the circle IMIN Ixx IMAX 2b x Ixx, Iyy Iyy • use geometry to find IMAX , IMIN and b x Ixy +Ixy (Ixx+Iyy)/2 Mohr’s Circle for Principal Inertia(Second Moment of Area about any Axis) ANGLES IN MOHR’S CIRCLE ARE TWICE THOSE IN THE CROSS SECTION!!! Lecture 9

  13. Example Lecture 9

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