ME221Lecture 91 ME 221 Statics Lecture #9 Sections 9.1 – 9.6

ME221Lecture 92 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

ME221Lecture 93 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

ME221Lecture 94 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

ME221Lecture 95 Moments of Area Normally, we want the moment with respect to centriod axes Moment of inertia –Moment about other axes derived from centroid case dA y x r Characterize distribution of area about the centroid

ME221Lecture 96 Moments of Basic Shapes Rectangle x y Circular x y dA = r dr d 

ME221Lecture 97 Polar Moment The polar moment is the second moment of area about the z-axis x y r Note that: I xx + I yy = J Oz

ME221Lecture 98 Parallel Axis Theorem MUST The centroid of the area MUST be one of the axes used in the parallel axis theorem. y dydy x x’ C

ME221Lecture 99 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

ME221Lecture 910 Principal Second Moments Definition of product moment of inertia Definition of principal axes –Product of inertia axis theorem Mohr’s circle to find principal axes Example

ME221Lecture 911 Product Moment of Inertia (Measures Antisymmetry) Basic section with two axes of symmetry x y Composite sections - product parallel axis theorem

ME221Lecture 912 Mohr’s Circle for Principal Inertia (Second Moment of Area about any Axis) +I xy -I xy I xx, I yy I yy x I xx I xy x draw point (I xx, I xy ) and the circle (I xx +I yy )/2 draw circle center (I xx + I yy )/2 x I MAX I MIN 22 use geometry to find I MAX, I MIN and  ANGLES IN MOHR’S CIRCLE ARE TWICE THOSE IN THE CROSS SECTION!!!

ME221Lecture 913 Example