2D Elements

We've mastered 1D bar elements. Now we extend to 2D elements that can model plates, shells, and plane structures. The fundamental concepts remain the same — shape functions, B-matrix, stiffness matrix — but with more DOFs and richer behavior.

Plane Stress vs Plane Strain

Before diving into elements, we need to understand the two 2D approximations:

Plane Stress

Used for thin structures where stress through thickness is negligible:

• Thin plates loaded in-plane
• Sheet metal
• Membranes

$$\sigma_z = \tau_{xz} = \tau_{yz} = 0$$

The constitutive matrix:

$$[D] = \frac{E}{1-\nu^2} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1-\nu}{2} \end{bmatrix}$$

Plane Strain

Used for long/thick structures where strain through thickness is negligible:

• Dams
• Retaining walls
• Long pipes under internal pressure

$$\varepsilon_z = \gamma_{xz} = \gamma_{yz} = 0$$

The constitutive matrix:

$$[D] = \frac{E}{(1+\nu)(1-2\nu)} \begin{bmatrix} 1-\nu & \nu & 0 \\ \nu & 1-\nu & 0 \\ 0 & 0 & \frac{1-2\nu}{2} \end{bmatrix}$$

The 3-Node Triangle (CST)

The simplest 2D element is the Constant Strain Triangle (CST) — 3 nodes, 6 DOFs total (2 per node).

Shape Functions

For a triangle with vertices at $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$:

$$N_1 = \frac{1}{2A}[(x_2 y_3 - x_3 y_2) + (y_2 - y_3)x + (x_3 - x_2)y]$$

$$N_2 = \frac{1}{2A}[(x_3 y_1 - x_1 y_3) + (y_3 - y_1)x + (x_1 - x_3)y]$$

$$N_3 = \frac{1}{2A}[(x_1 y_2 - x_2 y_1) + (y_1 - y_2)x + (x_2 - x_1)y]$$

Where $A$ is the triangle area:

$$2A = (x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2))$$

These are also called area coordinates or barycentric coordinates.

Displacement Interpolation

$$u(x,y) = N_1 u_1 + N_2 u_2 + N_3 u_3$$

$$v(x,y) = N_1 v_1 + N_2 v_2 + N_3 v_3$$

In matrix form:

$$\begin{Bmatrix} u \\ v \end{Bmatrix} = \begin{bmatrix} N_1 & 0 & N_2 & 0 & N_3 & 0 \\ 0 & N_1 & 0 & N_2 & 0 & N_3 \end{bmatrix} \begin{Bmatrix} u_1 \\ v_1 \\ u_2 \\ v_2 \\ u_3 \\ v_3 \end{Bmatrix}$$

Strain-Displacement (B-Matrix)

For 2D plane problems:

$$\{\varepsilon\} = \begin{Bmatrix} \varepsilon_x \\ \varepsilon_y \\ \gamma_{xy} \end{Bmatrix} = \begin{Bmatrix} \frac{\partial u}{\partial x} \\ \frac{\partial v}{\partial y} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \end{Bmatrix}$$

The B-matrix for the CST:

$$[B] = \frac{1}{2A} \begin{bmatrix} y_2-y_3 & 0 & y_3-y_1 & 0 & y_1-y_2 & 0 \\ 0 & x_3-x_2 & 0 & x_1-x_3 & 0 & x_2-x_1 \\ x_3-x_2 & y_2-y_3 & x_1-x_3 & y_3-y_1 & x_2-x_1 & y_1-y_2 \end{bmatrix}$$

Stiffness Matrix

$$[K_e] = \int_A [B]^T [D] [B] \, t \, dA = [B]^T [D] [B] \cdot t \cdot A$$

Since $[B]$ is constant, the integral is trivial — just multiply by the area!

The result is a 6×6 symmetric matrix.

Limitations of CST

• Constant strain can't capture strain gradients
• Poor in bending — need many elements
• Overly stiff behavior (locking)

Rule of thumb: Use CST only for membrane-dominated problems or as a learning tool. For real analysis, use higher-order elements.

The 6-Node Triangle (LST)

The Linear Strain Triangle (LST) adds mid-side nodes for quadratic shape functions:

• 6 nodes, 12 DOFs
• Quadratic displacement field
• Linear strain variation

$$N_1 = \xi(2\xi - 1)$$

$$N_2 = \eta(2\eta - 1)$$

$$N_3 = \zeta(2\zeta - 1)$$

$$N_4 = 4\xi\eta$$

$$N_5 = 4\eta\zeta$$

$$N_6 = 4\zeta\xi$$

Where $\xi$, $\eta$, $\zeta$ are area coordinates with $\xi + \eta + \zeta = 1$.

• Much better accuracy than CST
• Can capture bending behavior
• Handles stress concentrations better

The 4-Node Quadrilateral (Q4)

The bilinear quadrilateral has 4 nodes at corners, 8 DOFs total.

Shape Functions (Natural Coordinates)

In natural coordinates $(\xi, \eta) \in [-1, 1] \times [-1, 1]$:

$$N_1 = \frac{1}{4}(1-\xi)(1-\eta)$$

$$N_2 = \frac{1}{4}(1+\xi)(1-\eta)$$

$$N_3 = \frac{1}{4}(1+\xi)(1+\eta)$$

$$N_4 = \frac{1}{4}(1-\xi)(1+\eta)$$

Isoparametric Mapping

The same shape functions map geometry:

$$x = \sum_{i=1}^{4} N_i(\xi,\eta) x_i$$

$$y = \sum_{i=1}^{4} N_i(\xi,\eta) y_i$$

Jacobian Matrix

The Jacobian relates natural to physical derivatives:

$$[J] = \begin{bmatrix} \frac{\partial x}{\partial \xi} & \frac{\partial y}{\partial \xi} \\ \frac{\partial x}{\partial \eta} & \frac{\partial y}{\partial \eta} \end{bmatrix}$$

Then:

$$\begin{Bmatrix} \frac{\partial N_i}{\partial x} \\ \frac{\partial N_i}{\partial y} \end{Bmatrix} = [J]^{-1} \begin{Bmatrix} \frac{\partial N_i}{\partial \xi} \\ \frac{\partial N_i}{\partial \eta} \end{Bmatrix}$$

B-Matrix for Q4

$$[B] = \begin{bmatrix} \frac{\partial N_1}{\partial x} & 0 & \frac{\partial N_2}{\partial x} & 0 & \cdots \\ 0 & \frac{\partial N_1}{\partial y} & 0 & \frac{\partial N_2}{\partial y} & \cdots \\ \frac{\partial N_1}{\partial y} & \frac{\partial N_1}{\partial x} & \frac{\partial N_2}{\partial y} & \frac{\partial N_2}{\partial x} & \cdots \end{bmatrix}$$

Stiffness Matrix

$$[K_e] = \int_{-1}^{1} \int_{-1}^{1} [B]^T [D] [B] \, t \, |J| \, d\xi \, d\eta$$

The $|J|$ (Jacobian determinant) accounts for the coordinate transformation.

The 8-Node Quadrilateral (Q8)

Adding mid-side nodes gives the serendipity quadrilateral:

• 8 nodes, 16 DOFs
• Quadratic displacement
• Curved edges possible

$$N_i = \frac{1}{4}(1+\xi_i\xi)(1+\eta_i\eta)(\xi_i\xi+\eta_i\eta-1)$$

$$N_i = \frac{1}{2}(1-\xi^2)(1+\eta_i\eta) \text{ for } \xi_i = 0$$

$$N_i = \frac{1}{2}(1+\xi_i\xi)(1-\eta^2) \text{ for } \eta_i = 0$$

• Excellent accuracy
• Can model curved boundaries
• Standard choice for structural FEA

Element Comparison

Assembling 2D Elements

Assembly follows the same rules as 1D, but with DOF mapping:

• $u_i$ at global DOF $2i - 1$
• $v_i$ at global DOF $2i$

Element 1 uses global DOFs: 1, 2, 3, 4, 7, 8

Element 2 uses global DOFs: 3, 4, 5, 6, 7, 8

Shared DOFs (3, 4, 7, 8) accumulate stiffness contributions.

Mesh Quality

Poor element shapes degrade accuracy:

Aspect Ratio

$$AR = \frac{\text{longest edge}}{\text{shortest edge}}$$

Keep $AR < 3$ for good results. High aspect ratio = directional bias.

Jacobian Quality

For quads, the Jacobian determinant should be positive everywhere:

• $|J| > 0$: Valid element
• $|J| = 0$: Degenerate (collapsed) element
• $|J| < 0$: Inverted element (invalid!)

Skewness

For quads, interior angles should be close to 90°:

• Ideal: All angles = 90°
• Acceptable: 45° < angle < 135°
• Poor: angle < 30° or angle > 150°

Common Mistakes

• Using CST for bending: Needs excessive mesh refinement
• Inverted elements: Check Jacobian warnings
• Poor aspect ratio: Refine mesh or use different topology
• Ignoring plane stress/strain: Wrong $[D]$ matrix = wrong results
• Forgetting thickness: 2D plane elements need thickness $t$

Key Takeaways

• Plane stress for thin structures ($\sigma_z = 0$); plane strain for thick ($\varepsilon_z = 0$)
• CST (3-node triangle): Constant strain, 6 DOFs, poor for bending
• Q4 (4-node quad): Bilinear, 8 DOFs, needs Gauss integration
• Higher-order elements (LST, Q8) capture strain gradients better
• Isoparametric mapping: Same shape functions for geometry and field
• Jacobian: Relates natural to physical coordinates; must be positive
• Mesh quality: Aspect ratio, skewness, and Jacobian affect accuracy
• Assembly: Same scatter operation, but with 2 DOFs per node

What's Next

The stiffness matrix for Q4 and higher elements requires numerical integration because $[B]$ varies with position. We'll learn about Gauss quadrature — the standard method for evaluating these integrals efficiently and accurately.