3-Node Triangular Element (TRIA3)
3-Node Triangular Element (TRIA3): Theoretical Foundations
CST Element — The Starting Point of FEM
The 3-node triangular element is also called CST (Constant Strain Triangle), right?
Yes. Because strain is constant within the element, it's called the "Constant Strain Triangle". It is the first element in FEM published by Turner, Clough, Martin, and Topp in 1956, and is the very history of the finite element method itself.
The first FEM element! But on the TET4 page, you said "Don't use first-order elements," didn't you?
CST is the 2D version of TET4. It has the same drawbacks — strain is constant and cannot represent stress gradients. It should not be used in practical work. However, it is the optimal element for understanding the principles of FEM.
Shape Functions
The shape functions for CST are expressed using area coordinates $L_1, L_2, L_3$:
The B matrix is constant and can be calculated exactly without numerical integration:
$A_e$ is the area of the triangle, $t$ is the plate thickness.
Just multiply the element area and plate thickness. The calculation is very simple.
It's ideal as an introduction to FEM programming. The exercise of assembling CST by hand calculation helps deeply understand the meaning of the stiffness matrix.
Limitations of CST
To summarize the limitations of CST:
- Cannot represent bending — Strain is constant, so stress gradients from bending do not appear.
- Slow convergence — Requires 5 to 10 times more elements than Q8 or TRIA6 to achieve accuracy.
- Stress is constant within the element — Contours become stepped, and stress concentrations cannot be accurately evaluated.
The conclusion "don't use it" is exactly the same as for TET4.
CST (2D) = TET4 (3D). Same problem, same solution (switch to second-order elements).
When It's Okay to Use CST
The only legitimate use is for education and understanding the fundamentals of FEM. Perform hand calculations with CST to experience assembling the stiffness matrix, applying boundary conditions, and solving simultaneous equations. This is a required subject for FEM engineers.
Summary
Let me organize the theory of CST (TRIA3).
Key points:
- The first FEM element (1956) — The historical starting point of the finite element method.
- Constant strain — Stress is constant within the element. Cannot represent stress gradients.
- Do not use in practical work — Use TRIA6 (second-order triangle) or Q8.
- Optimal for education — Can learn FEM principles through hand calculations.
- 2D version of TET4 — Same problem, same solution.
CST Formulation of TRIA3 Element
The 3-node triangular element (TRIA3), also known as the Constant Strain Triangle (CST), was published in 1956 by Turner, Clough, Martin, and Topp in the "Journal of the Aeronautical Sciences" as the first FEM element. Defined by area coordinates L1, L2, L3 (L1+L2+L3=1), its linear shape functions result in constant strain within the element. This simplest 2D element continues to be used for connection elements and educational purposes.
Computational Methods for 3-Node Triangular Element (TRIA3)
CST Implementation
Is CST implementation really that simple?
It's the simplest among FEM elements. The B matrix is a constant matrix calculated only from the three node coordinates:
Determined solely by the difference in node coordinates. No numerical integration needed.
Element area $A$ is:
If $A > 0$, nodes are counterclockwise; if $A < 0$, clockwise.
Element Names by Solver
| Solver | Element Name | Remarks |
|---|---|---|
| Nastran | CTRIA3 | Plate thickness specified with PSHELL |
| Abaqus | CPS3 (Plane Stress), CPE3 (Plane Strain) | The most basic elements |
| Ansys | PLANE182 (degenerated) | Degenerated form by collapsing one side of a 4-node element |
Doesn't Ansys have a dedicated TRIA3 element?
Ansys's PLANE182 is 4-node but can be degenerated to 3-node for use. However, this degenerated form only has accuracy equivalent to CST, so again, not recommended.
Appearance of CST in Automatic Meshing
Does CST appear in automatic meshing?
CST may be inserted in areas that cannot be made into quadrilaterals during Q4 automatic meshing. Particularly, triangles can occur during the Closeout step of the Paving method (automatic quadrilateral mesh generation).
Countermeasures:
- Use settings to minimize the number of triangles.
- Control so triangles do not enter areas of interest.
- If possible, switch to Q8 second-order mesh.
Summary
Let me organize the implementation and practical handling of CST.
Key points:
- B matrix is constant — Simplest implementation. An introduction to FEM programming.
- Element inversion when $A < 0$ — Check node order.
- Can be mixed in during automatic meshing — Control so they don't enter areas of interest.
- Conclusion: Convert to TRIA6 — Meshes containing CST should be converted to TRIA6.
Closed-Form Stiffness Matrix of TRIA3
The stiffness matrix of TRIA3 can be calculated in closed form as K=t×A×B^T×D×B (t: plate thickness, A: element area, B: strain-displacement matrix, D: elasticity matrix), requiring no Gaussian integration. Area A is found as half the determinant of the coordinate matrix. Equivalent to 1-point integration (centroid), thus avoiding discussions about integration accuracy. Due to this simplicity, it continued to be used as the main tool for aircraft skin analysis until the 1980s.