Last Updated on July 25, 2023 by Maged kamel
Area and Cg of a Trapezium.
Reference handbook 10.00 value for area and Cg at the x-direction.
To get the area and Cg for a Trapezium, we will divide the trapezium into a rectangle for which we deduct two triangles and consider taking the first moment of area about a vertical axis y passing by the external edge point about the external axes. This will give the x-bar of the Trapezium.
As for the Y bar, we will do the same process, but the first moment of the area will be taken about an external axis passing by the base which is then called the x-axis.
This is a list of the first moment of area for typical plain shapes. The trapezium data for area and Cg are shown in the fifth shape.
![FE reference handbook list of areas and Cg.](https://magedkamel.com/wp-content/uploads/2022/01/page-1-post-8-moment-of-area.jpg)
It is required to get the same data as obtained from the FE reference handbook as attached in the next slide image, for case #No.5.
Area and Cg of a Trapezium at the x-direction.
The Trapezium with base b and the upper top side is a and height h, is divided into the following shapes:
1-A rectangle of base b and a height of h, its area A1=b*h, its CG is apart from the y-axis by a distance x1= (b1+0.50*a).
2-A left triangle of base b1 and a height of h, its area A2=1/2*b1*h, its CG is apart from the y-axis by a distance x2=(2/3*b1).
3-A right triangle of base b2 and a height of h, its area A3=1/2*b2*h, its CG is apart from the y-axis by a distance x3=(b1+1/3*b1).
The area of the trapezium is the area of the rectangle-Area of the left triangle- and area of the right triangle, The details of these shapes are shown in the next slide images.
![Area and Cg of a Trapezium-step-1.](https://magedkamel.com/wp-content/uploads/2022/01/page-2-post-8-moment-of-area.jpg)
The first moment of areas for these three shapes will be = the first moment of area for the Trapezium.
![The detailed shapes of areas and Cg distances.](https://magedkamel.com/wp-content/uploads/2022/01/page-3-post-8-moment-of-area.jpg)
We will proceed with our estimation by equating the first moment of area for the trapezium about the y-axis by the equivalent moments of areas for area A1, A2, A3, (-) minus sign is introduced since we are subtracting areas A2, A3 from A1.
![First moment of area about Y-axis.](https://magedkamel.com/wp-content/uploads/2022/01/page-4-post-8-moment-of-area.jpg)
The next two slide images give the full details of the estimation.
![The First moment of area about Y-axis.](https://magedkamel.com/wp-content/uploads/2022/01/page-5-post-8-moment-of-area.jpg)
Area and Cg of a Trapezium at the x-direction.
We will have the final expression for the distance of the Cg of the trapezium about the y-axis, which is the X-bar.
![X- bar value for a Trapezium.](https://magedkamel.com/wp-content/uploads/2022/01/page-6-post-8-moment-of-area.jpg)
The next link will show the same expression but considered as Cx, I have used b1 in lieu of C. when the Cx value is equal to 1/2*(b-a), where a is the upper side, while b is the lower side, then the Cg distance will be equal to b/2.
Area and Cg of a Trapezium at the y-direction.
The Trapezium with base b and the upper top side is a and height h, is divided into the following shapes:
1-A rectangle of base b and a height of h, its area A1=b*h, it is CG is apart from the x-axis by a distance y1= (1/2*h).
2-A left triangle of base b1 and a height of h, its area A2=1/2*b1*h, it is CG is apart from the y-axis by a distance y2=(2/3*h).
![The area and Cg of a Trapezium.](https://magedkamel.com/wp-content/uploads/2022/01/page-7-post-8-moment-of-area.jpg)
3-A right triangle of base b2 and a height of h, its area A3=1/2*b2*h, it is CG is apart from the x-axis by a distance y3=(2/3*h).
We divide by the area then Y-bar=(1/3)*(2a+b)/(a+b), this result matches the result obtained from the FE reference handbook table of areas and CG.
The final expression of the first moment of area about the x-axis, or A*y bar can be obtained and=(h^2/6)*(2a1+b).
This is a link for the pdf file used in the illustration of this post.
This is a link to review different types of areas and their CG: useful link for areas and Cg.
The second moment of area or the moment of inertia for a Trapezium can be found in post 19.
This is a link for the next post in the area and Cg of a parallelogram.