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Unformatted text preview: 1 CIV100 – Mechanics Module 4: Centroid and Moments of Inertia by: Jinyue Zhang 2011/10/13 1 Module Objective • By the end of this Module you should be able to: – Understand the underlying concept of centroids – Understand how moments relate to the concept of centroid – Know how to find centroids of composite shapes – Know how to calculate the fluid pressure – Understand the underlying concept of moments of inertia – Know how to calculate the moment of inertia for composite areas – Know how to use the parallel axis theorem 2011/10/13 2 Today’s Objective • Understand the concept of centroids • Be clear about the shapes you need to know the location of centroid • Be able to find the centroid of composite areas – Examples 2011/10/13 3 Center of Gravity and Centroid • Center of gravity – A point which locates the weight of a body – The balance-point of a body • Centroid – Geometric center of an object (an area or a volume) • The centroid coincides the center of gravity for a uniform or homogeneous body – The density or specific weight is constant throughout the body • Symmetric area/volume – Centroid must be on the line of symmetry! 2011/10/13 4 Areas You Need to Know x y H L H/3 L/3 x y H L H/3 L isosceles triangle x y H L H/2 L/2 x y x y 4r/3 π y x 4r/3 π 4r/3 π 2011/10/13 5 Volumes You Need to Know • All you need to know is prismatic volumes ! – You need to find the centroid of a cross section – Then draw a line passing through the centroid of that cross section and parallel to the z-axis (the height) – The centroid of this prismatic volume is on this line and at the half height! H H/2 H H/2 2 2011/10/13 6 Centroid of a Composite Area • Finding the centroid of a composite area is similar to find the equivalent force system – Divide the composite area to simpler shapes with known centroids – Choose any reference x-y coordinate system ( be smart ) – Locate centroid for each simple shape that you know the location of centroid – Locate the centroid of composite area by two variables (x, y) – Imagine each shape has a weight of 1N per unit area (for example 1N/m 2 ) – Calculate the sum of moments of all weights about x and y axes – The total weight of composite area should create the same moment about x and y axes respectively, then you can solve coordinates (x, y) of the centroid We are balancing the moment of areas at centroid, as such centroid is also called the moment of area! 2011/10/13 7 Example 1 • Locate the centroid of the plate. x y s 2011/10/13 8 Example 1 • Locate the centroid of the plate. x y Please note: The imagination of having a weight of 1N per unit area is only to demonstrate the method of finding centroid of composite areas. As this weight will be cancelled when calculating the moment of original divided system about x and y axes and the moment of equivalent system (the weight acting at the centroid of composite area) about same x and y axes respectively, so we only need...
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This note was uploaded on 12/10/2011 for the course CIV 100 taught by Professor Nahrvar during the Spring '08 term at University of Toronto.
- Spring '08