# In the Finite Element Method, which expression is correct for a linear triangular element if S is the shape function, Ae is its area, and K is a constant?

In the Finite Element Method, which expression is correct for a linear triangular element if S is the shape function, Ae is its area, and K is a constant?

(a) $\frac{\partial S}{\partial x}=\frac{K}{A_e}$

(b) $\frac{\partial S}{\partial y}=\frac{K}{A_e^2}$

(c) $\frac{\partial S}{\partial x}$=KAe

(d) $\frac{\partial S}{\partial y}$=KAe^2

This question was addressed to me in an online quiz.

The origin of the question is Plane Elasticity in section Plane Elasticity of Finite Element Method

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Right option is (a) $\frac{\partial S}{\partial x}=\frac{K}{A_e}$

Explanation: For a linear triangular (i.e., constant-strain triangle) element, the shape function    $\psi_i^e$ and its derivatives are given by $\psi_i^e=\frac{1}{2A_e}(\alpha_i^e+\beta_i^e x+\gamma_i^e y)$, $\frac{\partial \psi_i^e}{\partial x}=\frac{\beta_i^e}{2A_e}$ and $\frac{\partial \psi_i^e}{\partial y}=\frac{\gamma_i^e}{2A_e}$where Ae is the area of the element, α, β and γ are constants. Note that the derivatives of the shape function are constants.

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