3節点三角形要素(TRIA3)
Theory and Physics
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 $L_1, L_2, L_3$:
$$ N_i = L_i \quad (i = 1, 2, 3) $$
$$ u(x,y) = L_1 u_1 + L_2 u_2 + L_3 u_3 $$
The B matrix is constant and can be calculated exactly without numerical integration:
$$ [K_e] = A_e \cdot t \cdot [B]^T [D] [B] $$
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.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward during sudden braking? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, which assumes "forces are applied slowly enough that acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That is Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more under the same force? Obviously the rubber. This "resistance to stretching" is Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure" — they are different concepts.
- External force term (load term): Body force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire contents" (body force), the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression" — sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibration energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is important.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only equilibrium between external and internal forces.
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation). |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system. |
Numerical Methods and Implementation
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.
Linear Elements (First-Order Elements)
Linear interpolation between nodes. Computational cost is low, but stress accuracy is low. Beware of shear locking (mitigated by reduced integration or B-bar method).
Second-Order Elements (with Mid-Side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommendation: When stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Meshing
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix each iteration. Achieves quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Instead of applying full load at once, apply in small increments. The arc-length method (Riks method) can trace beyond extremum points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper" — reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer" — starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: opening to an estimated page and adjusting forward/backward (iterative) is more efficient than searching sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
First-order elements are like "approximating a curve with a ruler" — represented by straight line segments, so accuracy is limited. Second-order elements are like "using a flexible
Related Topics
なった
詳しく
報告