Derive shape functions of CST element

Consider a triangle with nodes 1, 2, 3 at $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$ numbered anticlockwise. Each node has two displacements $(u_i, v_i)$, so the element has 6 degrees of freedom.

         3 (x3,y3)
        / \
       /   \
      /     \
     1-------2

Displacement field

Choose a complete linear polynomial (from Pascal's triangle):

$$u = \alpha_1 + \alpha_2x + \alpha_3y,\qquad v = \alpha_4 + \alpha_5x + \alpha_6y$$

Apply the nodal conditions for $u$:

$$\begin{Bmatrix} u_1 \\ u_2 \\ u_3 \end{Bmatrix} = \begin{bmatrix} 1 & x_1 & y_1 \\ 1 & x_2 & y_2 \\ 1 & x_3 & y_3 \end{bmatrix}\begin{Bmatrix} \alpha_1 \\ \alpha_2 \\ \alpha_3 \end{Bmatrix}$$

Solving by Cramer's rule, where the determinant of the matrix equals $2A$ ($A$ = area of triangle):

$$u = N_1u_1 + N_2u_2 + N_3u_3,\qquad N_i = \frac{1}{2A}\left(a_i + b_ix + c_iy\right)$$

with

$$\begin{aligned} a_1 &= x_2y_3 - x_3y_2, & b_1 &= y_2 - y_3, & c_1 &= x_3 - x_2 \\ a_2 &= x_3y_1 - x_1y_3, & b_2 &= y_3 - y_1, & c_2 &= x_1 - x_3 \\ a_3 &= x_1y_2 - x_2y_1, & b_3 &= y_1 - y_2, & c_3 &= x_2 - x_1 \end{aligned}$$ $$2A = x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)$$

The same $N_i$ apply to $v$, so

$$\begin{Bmatrix} u \\ v \end{Bmatrix} = \begin{bmatrix} N_1 & 0 & N_2 & 0 & N_3 & 0 \\ 0 & N_1 & 0 & N_2 & 0 & N_3 \end{bmatrix}\{d\}$$

Properties

• $N_i = 1$ at node $i$ and $0$ at the other two nodes.
• $N_1 + N_2 + N_3 = 1$ at every point (and $\sum N_ix_i = x$, $\sum N_iy_i = y$).
• $N_i$ varies linearly along each edge and depends only on the two nodes of that edge, so displacements are continuous between adjacent elements.
• $N_i$ equal the area coordinates $L_i = A_i/A$.

Why constant strain

The strains are

$$\varepsilon_x = \frac{\partial u}{\partial x} = \frac{1}{2A}\sum b_iu_i,\quad \varepsilon_y = \frac{1}{2A}\sum c_iv_i,\quad \gamma_{xy} = \frac{1}{2A}\sum (c_iu_i + b_iv_i)$$

$b_i$ and $c_i$ depend only on the nodal coordinates, not on $x$ or $y$. So the strains (and stresses) are the same everywhere inside the element.