Constant-Strain Triangular Element Formulation

TL;DR

The Constant-Strain Triangle (CST) is the simplest 2D finite element, assuming constant strain and stress within itself. Its formulation involves defining displacement fields, calculating strains from these, and then relating strains to stresses using material properties. This element is fundamental for understanding more complex elements in Finite Element Analysis (FEA).

1. The Mental Model

Imagine you're trying to figure out how a flat, thin object deforms under a load. Instead of analyzing the whole thing at once, you break it into tiny triangles. For each triangle, you assume that the stretching and squishing (strain) inside it is perfectly uniform everywhere.

2. The Core Material

The Constant-Strain Triangular (CST) element is a fundamental building block in 2D FEA. It's a three-node triangular element where the displacement within the element is assumed to vary linearly. This linear displacement assumption directly leads to constant strain (and thus constant stress) throughout the element.

Here's how the formulation generally works:

Nodal Displacements

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Each node of the triangle has two degrees of freedom: displacement in the x-direction ($u$) and displacement in the y-direction ($v$).
Let the nodal displacements for the three nodes (1, 2, 3) be:
Node 1: ($u_1, v_1$)
Node 2: ($u_2, v_2$)
Node 3: ($u_3, v_3$)

The vector of nodal displacements for the element, $\{q\}$, is:
$\{q\} = [u_1, v_1, u_2, v_2, u_3, v_3]^T$

Displacement Field within the Element

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The displacement at any point $(x, y)$ within the element, $\{u(x,y)\} = [u(x,y), v(x,y)]^T$, is interpolated from the nodal displacements using shape functions ($N_i$). For a CST element, these shape functions are linear.

$u(x,y) = N_1 u_1 + N_2 u_2 + N_3 u_3$
$v(x,y) = N_1 v_1 + N_2 v_2 + N_3 v_3$

The shape functions $N_i$ are functions of $x$ and $y$, and they are 1 at node $i$ and 0 at the other nodes. They can be expressed in terms of the nodal coordinates $(x_i, y_i)$ and the element's area ($A_e$).

Strain-Displacement Relationship

Strain is the change in deformation. For 2D plane stress or plane strain problems, we're interested in normal strains ($\epsilon_x, \epsilon_y$) and shear strain ($\gamma_{xy}$).
The strain vector is $\{\epsilon\} = [\epsilon_x, \epsilon_y, \gamma_{xy}]^T$.

These strains are related to the displacements by:
$\epsilon_x = \frac{\partial u}{\partial x}$
$\epsilon_y = \frac{\partial v}{\partial y}$
$\gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}$

Since $u$ and $v$ are linear functions of $x$ and $y$, their derivatives with respect to $x$ and $y$ will be constants. This means the strain components $\epsilon_x, \epsilon_y, \gamma_{xy}$ are constant throughout the element, hence "constant-strain triangle."

We can write this relationship in matrix form as:
$\{\epsilon\} = [B]\{q\}$
Where $[B]$ is the strain-displacement matrix. For a CST element, the $[B]$ matrix is a $3 \times 6$ constant matrix (doesn't depend on $x,y$).

Constitutive Relationship (Stress-Strain)

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This relationship connects stress to strain using the material properties. For an isotropic linear elastic material, this is given by:
$\{\sigma\} = [D]\{\epsilon\}$
Where $\{\sigma\} = [\sigma_x, \sigma_y, \tau_{xy}]^T$ is the stress vector, and $[D]$ is the constitutive matrix (or elasticity matrix). The form of $[D]$ depends on whether it's a plane stress or plane strain problem.

Element Stiffness Matrix

The element stiffness matrix $[k]$ relates nodal forces to nodal displacements:
$\{f\} = [k]\{q\}$

For a CST element, the stiffness matrix can be derived using the principle of virtual work or potential energy:
$[k] = A_e t [B]^T [D] [B]$
Where $A_e$ is the area of the element and $t$ is its thickness.

This $[k]$ matrix is a $6 \times 6$ symmetric matrix unique to each element.

graph TD
    A["Define Nodal Coordinates (x_i, y_i)"] --> B["Calculate Element Area (A_e)"];
    B --> C["Formulate Shape Functions (N_i)"];
    C --> D["Derive Displacement Field (u, v)"];
    D --> E["Calculate Strain-Displacement Matrix ([B])"];
    E --> F{"Is it Plane Stress or Plane Strain?"};
    F -- Plane Stress --> G1["Formulate Plane Stress Constitutive Matrix ([D])"];
    F -- Plane Strain --> G2["Formulate Plane Strain Constitutive Matrix ([D])"];
    G1 --> H["Calculate Element Stiffness Matrix ([k])"];
    G2 --> H;
    H --> I["Assemble Global Stiffness Matrix and Solve"];

3. Worked Example

Let's find the $[B]$ matrix for a CST element.
Assume a triangular element with nodes 1, 2, 3 at coordinates:
Node 1: $(x_1, y_1)$
Node 2: $(x_2, y_2)$
Node 3: $(x_3, y_3)$

The displacement field is:
$u(x,y) = N_1 u_1 + N_2 u_2 + N_3 u_3$
$v(x,y) = N_1 v_1 + N_2 v_2 + N_3 v_3$

Where $N_i = \frac{1}{2A_e} (a_i + b_i x + c_i y)$
$a_1 = x_2 y_3 - x_3 y_2$
$b_1 = y_2 - y_3$
$c_1 = x_3 - x_2$
(And similarly for $a_2, b_2, c_2, a_3, b_3, c_3$ by cyclic permutation of indices).

Now, let's get the strain components:
$\epsilon_x = \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(N_1 u_1 + N_2 u_2 + N_3 u_3) = \frac{\partial N_1}{\partial x} u_1 + \frac{\partial N_2}{\partial x} u_2 + \frac{\partial N_3}{\partial x} u_3$
Since $\frac{\partial N_i}{\partial x} = \frac{b_i}{2A_e}$,
$\epsilon_x = \frac{1}{2A_e} (b_1 u_1 + b_2 u_2 + b_3 u_3)$

Similarly,
$\epsilon_y = \frac{\partial v}{\partial y} = \frac{1}{2A_e} (c_1 v_1 + c_2 v_2 + c_3 v_3)$

$\gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} = \frac{1}{2A_e} (c_1 u_1 + c_2 u_2 + c_3 u_3) + \frac{1}{2A_e} (b_1 v_1 + b_2 v_2 + b_3 v_3)$
$\gamma_{xy} = \frac{1}{2A_e} (c_1 u_1 + b_1 v_1 + c_2 u_2 + b_2 v_2 + c_3 u_3 + b_3 v_3)$

Now, we can assemble the $[B]$ matrix such that $\{\epsilon\} = [B]\{q\}$:
$
\begin{pmatrix}
\epsilon_x \
\epsilon_y \
\gamma_{xy}
\end{pmatrix}
= \frac{1}{2A_e}
\begin{pmatrix}
b_1 & 0 & b_2 & 0 & b_3 & 0 \
0 & c_1 & 0 & c_2 & 0 & c_3 \
c_1 & b_1 & c_2 & b_2 & c_3 & b_3
\end{pmatrix}
\begin{pmatrix}
u_1 \ v_1 \ u_2 \ v_2 \ u_3 \ v_3
\end{pmatrix}
$
So, the $[B]$ matrix is:
$[B] = \frac{1}{2A_e}
\begin{pmatrix}
b_1 & 0 & b_2 & 0 & b_3 & 0 \
0 & c_1 & 0 & c_2 & 0 & c_3 \
c_1 & b_1 & c_2 & b_2 & c_3 & b_3
\end{pmatrix}
$
This matrix is constant for any given element, demonstrating the "constant strain" nature.

4. Key Takeaways

• The CST element assumes linear displacement variation within the element, leading to constant strain and stress.
• The strain-displacement matrix $[B]$ for a CST element is constant, which simplifies calculations compared to higher-order elements.
• The element stiffness matrix $[k]$ directly links nodal forces to nodal displacements, and it's calculated using the element's area, thickness, and $[B]$ and $[D]$ matrices.
• The choice between plane stress and plane strain conditions dictates the form of the constitutive matrix $[D]$.
• CST elements are computationally inexpensive but can be "too stiff" and require a fine mesh to accurately capture complex stress gradients.
• They are excellent for introductory FEA and understanding the basics of element formulation.

Common Mistakes:
- Forgetting to correctly calculate the element area $A_e$, especially when using nodal coordinates.
- Mixing up plane stress and plane strain constitutive matrices, which changes the material's response.
- Incorrectly deriving the $b_i$ and $c_i$ terms for the shape functions; always ensure cyclic permutation for accuracy.
- Assuming the $[B]$ matrix depends on position $(x,y)$ within a CST element; it's explicitly constant.

5. Now Try It

Given a CST element with nodes at (0,0), (2,0), and (0,3), calculate the values for $a_1, b_1, c_1$ and then construct the $[B]$ matrix for this specific element. Assume the thickness is not relevant for the $[B]$ matrix itself, only for $[k]$. What success looks like: a correctly calculated $[B]$ matrix with numerical values for each term.