4節点四辺形要素(QUAD4)
Theory and Physics
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.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, meaning "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? 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, based on the assumption that "acceleration can be ignored because forces are applied slowly". 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", right? That's Hooke's law $F=kx$, and it's the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more when pulled with the same force? Obviously the rubber. This "resistance to stretching" is the 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 forces $f_b$ (gravity, etc.) and surface forces $f_s$ (pressure, contact forces, etc.). Think of it this way — the weight of a truck on a bridge is a force "acting on the entire volume" (body force), while 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 ending up with "compression" — it 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 the vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — they intentionally absorb 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 crucial.
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 the 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 the balance 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 inconsistencies 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 to N in mm system, N in m system. |
Numerical Methods and Implementation
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.