ME 221Lecture 181 ME 221 Statics Lecture #18 Sections 9.1 – 9.6

ME 221Lecture 182 Homework #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 17

ME 221Lecture 183 Exam #2 Results Average = xx.x Scores posted on Angel Solution will be posted on Angel today See syllabus for regrade policy Last day to drop without a grade reported is: Tuesday, October 14

ME 221Lecture 184 Second 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 theorem

ME 221Lecture 185 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

ME 221Lecture 186 Moments of Basic Shapes Rectangle x y Circular x y dA = r dr d 

ME 221Lecture 187 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

ME 221Lecture 188 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

ME 221Lecture 189 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

ME 221Lecture 1810 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

ME 221Lecture 1811 Product Moment of Inertia Basic section with two axes of symmetry x y Composite sections - product parallel axis theorem

ME 221Lecture 1812 Mohr’s Circle for Principal Inertia +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!!!

ME 221Lecture 1813 Example