14.3 CENTROIDS

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This is also known as the geometric centre, or first moment of area.

The fundamental definition for Centroids is given below, although a more efficient method for problems with simple geometries is discussed later.

When finding centroids, look for symmetry, the centroids will be in the center of symmetrical sections.

A simple example of application for these relationships is given below,

Lets try a more practical problem using the centroid, ([Hibbeler, 1992], prob. 9-16, pg. )

Consider the example below,

14.3.1 Finding Centroids Using Composite Shapes

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This method basically uses tables, or obvious centroids for basic shapes in a complex shape. By using a summation of centroids, weighted by areas/volumes divided by total area.

A simple example for a two dimensional problem is given below, ([Hibbeler, 1992], prob. 9-48, pg. )

Next, lets consider a problem using the composite method in 3D volumes, ([Hibbeler, 1992], prob 9-68, pg. )

Consider the example below,

We will often encounter distributed forces in real problems. For the convenience of calculation we want to simplify these to point forces.

Consider the example below,

An example of a roughly distributed load can be seen below,

[picture]

14.3.2 Pappus and Guldinus - Rotated Sections

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If we take some curve and rotate it about some axis, it will sweep out a shell of some volume. We refer to this as a Volume of Revolution.

We can calculate the surface area of these shapes using the centroid and the arc length of the curve,

We can easily calculate the volume of these shapes using the centroid and the area under the curve,

These methods can be of particular interest when the shapes are difficult to integrate in three dimensions, or when we start with a profile of a shape. One example of this would be a program to drive a CNC lathe.

Try the example below,

14.3.3 Practice Problems

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1. The plate below has a triangular hole. We are asked to select a base width `b' for the triangle so that the reaction at roller B is 200N. The plate material weighs 1Kg per square meter.

2. The 500 Kg plate below is to be used in an industrial machine. The basic shape is rectangular, but there is an arch cut in the bottom. Considering the centroid, and the applied force, determine the magnitude of the force (P) that is required to pull out the wedge if both sides of the wedge have a coefficient of friction of 0.2.

3. A thin flat plate is suspended by means of a screw eye and 2 cords, AC and BC, as depicted in the diagram below. Given that the plate has a mass of 30kg per square meter of surface area, determine the reaction force at point O and the tension forces in the cords.

4. Find the x-y centroid of the shape below which is essentially rectangular, except for the shape cut out of the bottom (the function is given).

5. What is the x-y centroid of the shape below if the plate is homogenous?

6. The mass below (a rectangle with a parabola cut out) is suspended by two cables with a diameter of 2mm. The cables have an undeflected length of 2m.

a) Find the x centroid for the plate and the force of gravity on the plate if the material weighs 50 kg per square meter.

b) Determine the new lengths of the cables at A and at B. Assume that the cables have E = 20GPa.

c) Based on the elongated cables, what angle does the plate sit at after it is released and deflects?