Rectangular Elements

While the triangular element is simple and can be used to mesh any two-dimensional geometry, its accuracy is limited by the assumption of constant strain. To improve upon this, we introduce the 4-node rectangular element. This element allows for a more complex strain distribution, leading to more accurate results.

1. Displacement Field

For this element, the displacement field (e.g., for the u-displacement) is approximated using a four-term polynomial. A crucial addition is the xy term, which allows for a non-constant strain distribution.

$$u(x,y)=a_1+a_{2}x+a_{3}y+a_{4}xy$$

This can be written in matrix form:

By enforcing this equation at each of the four nodes, we can solve for the coefficients a in terms of the nodal displacements u. This leads to the familiar relationship $u=𝐍𝐮$, where N is the vector of shape functions.

2. Shape Functions

Instead of a brute-force matrix inversion, the shape functions for a rectangular element can be constructed elegantly through the product of one-dimensional linear interpolation functions. Consider a rectangle with dimensions a and b. We can define simple linear functions in each direction:

• In the x-direction: $f_1(x)=1-\frac{x}{a}$ $f_2(x)=\frac{x}{a}$
• In the y-direction: $g_1(y)=1-\frac{y}{b}$ $g_2(y)=\frac{y}{b}$

The two-dimensional shape functions are then formed by taking products of these 1D functions. For a node i, the shape function is the product of the 1D functions that are equal to 1 at that node.

• $N_1(x,y)=f_1(x)g_1(y)=(1-\frac{x}{a})(1-\frac{y}{b})$
• $N_2(x,y)=f_2(x)g_1(y)=(\frac{x}{a})(1-\frac{y}{b})$
• $N_3(x,y)=f_2(x)g_2(y)=(\frac{x}{a})(\frac{y}{b})$
• $N_4(x,y)=f_1(x)g_2(y)=(1-\frac{x}{a})(\frac{y}{b})$

3. Strain-Displacement Matrix and Stiffness

Since the shape functions now contain x and y terms, their derivatives are no longer constant. For example, for $N_1$:

$$\frac{\partial N_1}{\partial x}=-\frac{1}{a}(1-\frac{y}{b}),\frac{\partial N_1}{\partial y}=-\frac{1}{b}(1-\frac{x}{a})$$

The B matrix will now contain terms that are functions of x and y. This means that the strain, given by $\{ϵ\}=𝐁𝐪$, is no longer constant within the element. It can vary linearly, which is a significant improvement over the Constant Strain Triangle.

The element stiffness matrix is calculated using the standard formula:

$$𝐊={\int }_V{𝐁}^T𝐄𝐁dV=t{\int }_0^b{\int }_0^a𝐁(x,y{)}^T𝐄𝐁(x,y)dxdy$$

Because the B matrix is a function of x and y, the integrand is no longer a constant and the integration must be performed explicitly.