Spring Element and Connector
Theory and Physics
What is a Spring Element?
Professor, when do we use the "spring element" in FEM?
The spring element is the simplest element that elastically connects two points. It is widely used not only for modeling physical springs (coil springs, etc.) but also as a simplified model for joints and support conditions.
What are some specific examples?
Spring Element Stiffness Matrix
The stiffness matrix for a linear spring is 2×2:
Here, $k$ is the spring constant. It is the simplest stiffness matrix among all FEM elements.
It has the same form as the truss part ($EA/L$) of a beam element.
A truss element is equivalent to a spring element with $k = EA/L$. The spring element is a generalization of this, allowing any $k$ to be set.
Types of Spring Elements
| Type | DOF | Application |
|---|---|---|
| Translational Spring (SPRING1/2) | 1 translational direction | Ground springs, axial coupling |
| Rotational Spring | 1 rotational direction | Semi-rigid connections |
| 6DOF Spring (BUSHING) | All 6 degrees of freedom | Bushings, elastic bearings |
| Grounded Spring | Only 1 node | Ground support, elastic support |
Is a grounded spring a spring where one side is fixed to the "ground"?
Yes. It's also called a 1-node spring. A typical example is representing a foundation on soil with springs of "ground reaction coefficient × area". In Nastran, it's CELAS1/CELAS2; in Abaqus, it's SPRING1.
Nonlinear Springs
Are there nonlinear springs?
Yes. The force-displacement relationship is defined by a table (piecewise linear).
- Bilinear spring — Elastoplastic spring with a yield point
- Gap spring — Force is generated only after a certain gap is exceeded
- Nonlinear elastic — Arbitrary F-δ curve
Abaqus's *CONNECTOR ELEMENT is a versatile connector that can freely define nonlinear force-displacement, moment-rotation, friction, and damping.
Element Names by Solver
| Type | Nastran | Abaqus | Ansys |
|---|---|---|---|
| Scalar Spring | CELAS1/2 | SPRING1/2 | COMBIN14 |
| Bushing | CBUSH | *CONNECTOR | COMBIN40 |
| Nonlinear Spring | CBUSH1D(NL) | *CONNECTOR(NL) | COMBIN39 |
Is Abaqus's CONNECTOR ELEMENT the most versatile?
Yes. Abaqus's *CONNECTOR can define "spring", "damper", "friction", "lock", and "stopper" in a single element. Nastran's CBUSH also handles multiple DOFs, but Abaqus is more flexible for nonlinear behavior.
Summary
Let me organize the theory of spring elements.
Key points:
- The simplest element that elastically connects two points — Stiffness matrix is 2×2
- Widely used for simplifying joints and support conditions — Bolts, ground springs, semi-rigid connections
- Nonlinear springs — Force-displacement table, gap, friction
- CONNECTOR (Abaqus) is the most versatile — Integrates multi-DOF, nonlinear, friction
- The validity of the spring constant governs the results — Set $k$ with a physical basis
The last point is crucial. If the spring constant is set arbitrarily, the results will also be arbitrary.
Spring elements are easy to set up, but the physical basis for the spring constant is everything. Ground reaction coefficient, bolt axial stiffness, joint rotational stiffness... Whether you can calculate these correctly is a measure of an engineer's skill.
Derivation of the Spring Element Stiffness Matrix
The stiffness matrix of a 1D spring element is k[1,-1;-1,1] (k: spring constant), representing the simplest finite element. This 2×2 matrix is derived directly from Hooke's law F=kδ and appeared in the seminal 1956 finite element method paper "Stiffness and Deflection Analysis of Complex Structures" by Turner, Clough, Martin, and Topp. It serves as the textbook starting point for all structural FEM elements.
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, based on the assumption that "forces are applied slowly enough to ignore acceleration". 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. So, a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. 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 vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they deliberately absorb vibrational 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 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
Spring Element Implementation Details
Are there any points to be careful about when implementing spring elements?
They seem simple but have unexpected pitfalls.
Coordinate System Issues
How do you define the direction of a spring?
A spring element's stiffness acts in a specific direction in the global coordinate system. To place a spring in a diagonal direction, you need to either define a local coordinate system or specify a direction vector.
Points to note:
- Nastran's CELAS1 specifies the direction (DOF number) in grid units
- Abaqus's SPRING element defaults to the direction between connected nodes but can also specify arbitrary directions
- Ansys's COMBIN14 defaults to global axis directions. Can be changed to local directions via KEYOPT
If the direction is wrong, the spring will act in an unintended direction.
It's the most common mistake. When you add a spring element but the results hardly change, the spring direction is often wrong.
Modeling Grounded Springs
How do you model ground springs (grounded springs)?
Let's take pile foundation ground springs as an example. Represent ground reaction in the depth direction with springs:
Here, $k_s$ is the ground reaction coefficient (kN/m³), $D$ is the pile diameter, and $\Delta z$ is the element length.
How do you determine the ground reaction coefficient?
It is determined from ground surveys (boring, standard penetration tests, etc.).
| Ground Type | Typical $k_s$ (kN/m³) |
|---|---|
| Soft Clay | 2,000 ~ 5,000 |
| Medium Clay | 10,000 ~ 30,000 |
| Hard Clay | 30,000 ~ 100,000 |
| Loose Sand | 5,000 ~ 15,000 |
| Dense Sand | 30,000 ~ 100,000 |
The order of magnitude of the spring constant varies by two digits! Accurate evaluation of the ground is crucial.
The uncertainty in ground spring constants directly affects structural response. Sensitivity analysis (varying $k_s$ up and down to see response changes) is essential.
CONNECTOR Element (Abaqus)
Please teach me how to set up Abaqus's CONNECTOR element.
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Please teach me how to set up Abaqus's CONNECTOR element.
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