Question

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

Answer 1

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.