Breif description of Post 7-inertia.

7- How to find Area and Cg for a rectangle?

Area and Cg for a rectangle.

Reference handbook 10.00 value for area and Cg at the x- direction.

To get the area and Cg for a rectangle, we will integrate an infinitesimal area of (dx) width and dy height, about the external axes .it is required to get the same data as obtained from the FE reference handbook as attached in the next slide image, where Xc=b/2. Xc, we call it X- bar.

You can click on any picture to enlarge, then press the small arrow at the right to review all the other images as a slide show.

Reference  handbook values of the area

This is a picture from the FE reference handbook for the area and Cg of the different plain shapes.

Area and Cg for a rectangle in the x- direction-(x-bar).

Our new topic will be the estimation of the area and Cg of a rectangle. The procedure will be done by integrating a strip of the area of(dx*dy), the summation of this tiny area will be equal to the total area.

Area and Cg for a rectangle by integration

Likewise, the summation of this area’s first moment at the Y-axis will give the first moment of the area of the whole rectangle at the y- axis.

How to do the integration for an area(dx)*(dy) for the rectangle area?

But the double integration will include two operations, the first operation will be the integration of dy from y=0 to y=h, thus a rectangular strip will be created, then the moment of the strip about y will give the final value of the first moment of area of the rectangle at the Y-axis.

The final moment of area about Y-axis for a rectangle.

The detailed calculation for the first moment of area for the rectangular shape is presented in the next slide image.
The final value of the first moment of the area at the y-axis will be h*b^2/2.

The final value of x- bar will be obtained by the division of the value of the first moment of the area by the value of the rectangular area.

The value of x bar for a rectangle.

The area of the rectangle will be the product of the base by the height, which is =b*h. The final value of x- bar=b^2*h/2/(0.50*b*h)=b/2, or half the breadth, the result will match the Reference handbook value given.

Reference handbook 10.00 value for area and Cg in the y- direction.

Reference handbook values of the area

To get the area and Cg for a rectangle, we will integrate an infinitesimal area of (dx) width and dy height, about the external axes. It is required to get the same data as obtained from the FE reference handbook as attached in the next slide image, where yc=h/2. yc we call it y- bar.

Area and Cg for a rectangle at the y- direction-(y-bar).

Now we will estimate the first moment of area about the x-axis. The procedure will be done by integrating a strip of the area of(dx*dy), the summation of this tiny area will be=the total area.

The process is done by using the double integration method, the full details are shown in the next slide images.

The first moment of area about the x-axis for a rectangular shape.

Likewise, the summation of this area’s first moment at the x-axis will give the first moment of the area of the whole rectangle at the x-axis which will be =b*h^2/2.


The final value of the y- bar will be obtained by the division of the value of the first moment of the area by the value of the rectangular area.
The area of the rectangle will be the product of the base by the height, which is =b*h. The final value of y- bar=h^2*b/2/(0.50*b*h)=h/2, or half the total height f the rectangle, the result will match the Reference handbook value given. Thus the final value of area and Cg for a rectangle is completed.
Another alternative is considering a vertical strip of breadth dx and height =h and performing an integration about the y-axis, it will be a single integration, and we will get the same result for x-bar.

The final y-bar value for a rectangle area.

Another alternative is considering the horizontal strip of breadth b and thickness =dx and performing an integration about the x-axis, it will be a single integration, we will get the same result for y-bar.

This is a link for the pdf file used in the illustration of this post.

For a good external reference, please refer to the Centroid of an Area by Integration.

The next post will be Area and Cg for a Trapezium.

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