4-Node Quadrilateral Element (QUAD4)
4-Node Quadrilateral Element (QUAD4): Theoretical Foundations
Q4 Element — The Mainstay of 2D FEM
Professor, the 4-node quadrilateral element (Q4) is the foundation of 2D FEM, right?
Yes. Q4 is the element that has been studied and refined the most in the history of FEM. It can be considered the 2D version of HEX8. It can be used for all cases: Plane Stress, Plane Strain, and axisymmetric problems.
Shape Functions
The shape functions for Q4 are bilinear in natural coordinates $(\xi, \eta)$:
$$ N_i = \frac{1}{4}(1 + \xi_i \xi)(1 + \eta_i \eta) $$
The shape functions for Q4 are bilinear in natural coordinates $(\xi, \eta)$:
The four nodes correspond to $(\xi, \eta) = (\pm 1, \pm 1)$.
Bilinear means four terms: $1, \xi, \eta, \xi\eta$. There's a cross term $\xi\eta$.
Thanks to this cross term, Q4 has higher bending representation capability than the 3-node triangle (CST). However, to represent complete bending deformation, $\xi^2, \eta^2$ terms are needed, which Q4 lacks.
Shear Locking Problem
Please tell me about shear locking in Q4.
When trying to represent pure bending with full integration (2×2 Gauss points) in Q4, parasitic shear strain occurs, causing the element to "lock up." This is shear locking.
Physically, it happens like this:
- In pure bending, the top surface stretches and the bottom surface contracts.
- When representing this deformation with Q4's bilinear shape functions, shear strain inevitably appears.
- The energy from this shear strain is stored unnecessarily, making the element overly stiff.
What are the countermeasures?
Three methods:
1. Reduced integration (1×1 Gauss point) — Evaluates shear at a single integration point. Locking disappears but hourglass modes occur.
2. Incompatible modes — Adds internal degrees of freedom corresponding to $\xi^2, \eta^2$. Used in Nastran's CQUAD4, Abaqus's CPS4I.
3. Assumed Natural Strain (ANS) — Assumes shear strain separately.
Nastran's CQUAD4 has incompatible modes enabled by default, right?
Yes. Nastran's CQUAD4 is considered one of the most successful elements in the history of FEM. It's the improved version published by MacNeal and Harder in 1985, which significantly improved bending accuracy with incompatible modes while also satisfying the patch test.
Q4 Strengths and Weaknesses
| Characteristic | Strengths | Weaknesses |
|---|---|---|
| Mesh Generation | Mapped meshing is easy | Free meshing often mixes in triangles |
| Accuracy | High accuracy with incompatible modes | Accuracy degrades with distorted shapes |
| Computational Cost | Low (8 DOF) | — |
| Curved Surface Approximation | — | Straight edges only (curves approximated by polylines) |
Summary
Let me organize the theory of Q4.
Key points:
- Bilinear shape functions — Includes cross term $\xi\eta$. More accurate than CST.
- Shear locking — Occurs with full integration. Countermeasures: reduced integration or incompatible modes.
- Nastran's CQUAD4 has built-in incompatible modes — One of the most successful elements in FEM history.
- 2D practical standard — Widely used for Plane Stress, Plane Strain, and axisymmetric problems.
- Weak against distorted shapes — Management of aspect ratio and skewness is important.
The concepts of locking and hourglassing I learned on the HEX8 page apply directly to Q4 as well, don't they?
Since Q4 is the 2D version of HEX8, exactly the same problems occur. If you understand them in 2D, extending to 3D is easy.
Q4 Element Isoparametric Formulation
The isoparametric formulation of the Q4 element was developed in the 1960s by Irons, Ergatoudis, and others at the University of Birmingham. By using natural coordinates (ξ, η), arbitrary quadrilateral shapes could be mapped to a unit square, enabling high-accuracy stiffness matrix calculations in combination with Gaussian integration. It is still adopted as a core element in commercial codes today.
Computational Methods for the 4-Node Quadrilateral Element (QUAD4)
Q4 Integration Scheme Comparison
Please tell me the characteristics of each integration scheme for Q4.
| Integration | Gauss Points | Shear Locking | Hourglass | Computational Cost |
|---|---|---|---|---|
| Full Integration (2×2) | 4 | Yes | No | Baseline |
| Reduced Integration (1×1) | 1 | No | 3 modes | Low |
| Incompatible Modes | 4 | No | No | Slightly higher |
3 hourglass modes means 3 out of 8 DOF are zero-energy?
Out of the 5 deformation modes (8 DOF minus 3 rigid body modes: 2 translation + 1 rotation), only 2 are evaluated with reduced integration. The remaining 3 are hourglass modes.
Element Names by Solver
| Variation | Nastran | Abaqus | Ansys |
|---|---|---|---|
| Standard Q4 | CQUAD4 | CPS4 (Plane Stress) | PLANE182 |
| Reduced Integration | — | CPS4R | PLANE182(red.) |
| Incompatible Modes | CQUAD4 (default) | CPS4I | PLANE182(EAS) |
| Hybrid | — | CPE4H (Plane Strain) | — |
Nastran's CQUAD4 has incompatible modes enabled by default, so no special settings are needed.
Nastran users enjoy a simple world of "just use CQUAD4 and you're good". In Abaqus or Ansys, users need to choose the integration scheme, which demands more knowledge.
Impact of Mesh Quality
Is Q4 sensitive to shape distortion?
Extremely sensitive. There is a significant accuracy difference between an ideal square Q4 and a distorted parallelogram Q4.