For a linear triangular element with (xi, yi) as the coordinates of the ith node of the element the area=10units, the value of ∑βi from the standard relation αi+βiX+γiY=(2/3)*Area where X=∑xi, Y=∑yi is ___

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For a linear triangular element with (xi, yi) as the coordinates of the ith node of the element the area=10units, the value of ∑βi from the standard relation αi+βiX+γiY=(2/3)*Area where X=∑xi, Y=∑yi is ___

(a) 0

(b) 10

(c) 20

(d) 30

The question was posed to me by my college director while I was bunking the class.

This intriguing question comes from Boundary Value Problems in portion Single Variable Problems of Finite Element Method

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Right choice is (a) 0

For explanation I would say: A linear triangular element has 3 nodes. With (xi, yi) as coordinates of ith node, the twice of area is given by determinant of the matrix $\begin{pmatrix}1&x1&y1\\1&x2&y2\\1&x3&y3&\end{pmatrix}$ which equals to (x1y2−x2y1)+(x2y3−x3y2)+(x3y1−x1y3). Then from the standard relation we have ∑βi=(y2−y3)+(y3−y1)+(y1−y2)

=y2−y3+y3−y1+y1–y2

=0.

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