Fatigue Evaluation of Welded Joints
Theory and Physics
Fatigue of Welded Joints
Professor, are welded joints weak against fatigue?
Welded joints have significantly lower fatigue life than the base material due to residual stress, stress concentration, and welding defects (blowholes, undercut). The majority of fatigue failures in steel structures occur at the weld toe.
Welding Fatigue Assessment Methods
| Method | Stress Definition | Characteristics |
|---|---|---|
| Nominal Stress Method | Average stress over the cross-section | Simplest. Design codes (EN 1993-1-9, IIW) |
| Hot Spot Stress Method | Structural stress at the weld toe | Extrapolation from FEM results. Mesh-insensitive |
| Notch Stress Method | Notch stress at the weld toe ($R_{ref} = 1$ mm) | Most detailed. FEM-dependent |
| Crack Propagation Method | Stress Intensity Factor | Life of existing cracks. Influence of welding defects |
The Hot Spot Stress Method is an intermediate approach.
Extrapolate FEM results from positions 0.4t and 1.0t ($t$: plate thickness) away from the weld toe to estimate the structural stress at the toe. Independent of mesh size (with consistent extrapolation rules). Recommended by IIW.
Summary
Liberty Ship Welding Cracks and Fatigue
During World War II, serious cracks occurred in about 400 out of 2710 American Liberty ships, with dozens breaking in half at sea. One cause was the combination of fatigue at welded joints and low-temperature brittleness. Subsequent research revealed that stress concentration factors Kt=2~3 occur at weld bead toe ends, which became the origin of the IIW welding fatigue design standards.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "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, assuming "forces are applied slowly so 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's 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. 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"—different concepts.
- External Force Term (Load Term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (pressure, contact forces). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire contents" (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 it becomes "compression"—sounds like a joke, but it actually happens when coordinate systems rotate 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—they intentionally absorb vibration energy for a smoother ride. If damping were zero? Buildings would keep swaying 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, with linear stress-strain relationship.
- Isotropic Material (unless specified otherwise): 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: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extensions are needed.
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. Beware 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
FEM for Welding Fatigue
FEM procedure for the Hot Spot Stress Method:
1. Model the welded structure with shell elements (mid-surface of plate thickness)
2. Read stresses at positions 0.4t and 1.0t from the weld toe
3. Estimate structural stress at the toe by linear extrapolation
4. Perform life assessment using IIW FAT classification S-N curves
Notch Stress Method
IIW Effective Notch Stress Method: Assign a virtual notch radius of $R_{ref} = 1$ mm (for steel) to the weld toe and weld root, calculate the notch stress via FEM. Perform unified assessment with the FAT225 S-N curve.
Summary
How to Use the Hot Spot Stress Method
Widely used in welding fatigue assessment, the Hot Spot Stress Method obtains the hot spot stress by linearly extrapolating surface stresses from two points at 0.4t and 1.0t (t is plate thickness) from the weld toe. This stress is compared with S-N curves from IIW-specified FAT classes (e.g., FAT90). Since the extrapolation interval setting greatly affects results, a mesh size of t/4 or less is recommended.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with Mid-Side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2~3 times. Recommended when stress evaluation is critical.
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 Mesh
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. Quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Lower cost per iteration, but linear convergence speed.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$〜$10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Apply full load not all at once, but in small increments. The arc-length method (Riks method) can track 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 accuracy improves with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to estimate where to open it and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Welding Fatigue in Practice
Essential for steel structures like bridges, ships, cranes, pressure vessels, and offshore structures.
Examples of IIW FAT Classification
| Joint Type | FAT (N/mm²) |
|---|---|
| Base Material (ground surface) | FAT 160 |
| Butt Weld (weld reinforcement removed) | FAT 112 |
| Butt Weld (with weld reinforcement) | FAT 90 |
| Fillet Weld (cruciform joint) | FAT 71 |
| Non-Load Bearing Fillet Weld | FAT 80 |
FAT = Stress range $\Delta\sigma$ causing failure at 2×10⁶ cycles.
Practical Checklist
20-Year Life Verification for Ship Hull Welding Fatigue
DNVGL (Det Norske Veritas Germanischer Lloyd) design rules require assessing the fatigue life of ship hull welded joints for 20 years. Under North Sea wave spectrum (Hs=3m) loading cycles reach about 10⁸ over 20 years, making Δσ=71MPa the fatigue limit for FAT71 class welds. The FEM direct hot spot assessment method, popular since the 2000s, has significantly improved assessment accuracy.
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer will be far from reality. Confirm that results stabilize across at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct".
Thinking About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling real-world constraint conditions is actually the most critical step in the entire analysis.
Software Comparison
Welding Fatigue Tools
Selection Guide
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