Welding Residual Stress Analysis
Theory: how residual stress arises
Overview
Why does welding leave residual stress when no external force is applied?
Because it heats and cools locally. The weld is heated to well over a thousand degrees and tries to expand, but cooler surroundings hold it back, so the hot, weakened weld deforms plastically in compression. On cooling it now wants to be shorter than before, is held by the surroundings, and is left in tension. The result is tensile residual stress near yield along the weld and balancing compression further out. This stress reduces fatigue strength and resistance to stress corrosion cracking, and appears as distortion. Analysis solves the temperature history with a moving heat source, then the resulting thermal strain in an elasto-plastic analysis with temperature-dependent Young's modulus, yield stress and expansion coefficient.
Cooling time t8/5 (Rosenthal)
q/v is net heat input per unit length, k thermal conductivity, ρc volumetric heat capacity, d plate thickness and T₀ preheat temperature. Thick plate (3D heat flow) and thin plate (uniform through thickness) use different formulas; below the thickness where they coincide, use the thin-plate one.
Yielding by restraint
The temperature change at which fully restrained material reaches yield; σ_y is yield stress, E Young's modulus and α the expansion coefficient.
The winter welded ships cracked
Liberty ships, welded cargo vessels mass-produced in the United States during the Second World War, suffered repeated hull fractures, notably in the cold North Atlantic; some broke in two while moored. The causes combined steel that turned brittle at low temperature, hatch corners that concentrated stress, and tensile residual stress from welding. These experiences made steel toughness specifications, weld design and the assessment and relief (stress-relief annealing) of residual stress major engineering concerns.
Worked examples
Example 1: cooling time for an 800 J/mm arc weld
25 V, 200 A, 5 mm/s travel and arc efficiency 0.8 give a net heat input of 800 J/mm. Steel: conductivity 30 W/(m·K), volumetric heat capacity 4.8 MJ/(m³·K). Thick- and thin-plate formulas coincide at 16.7 mm.
| Thickness | Formula | t8/5 (no preheat) | t8/5 (150 °C preheat) |
|---|---|---|---|
| 6 mm | Thin | 26.5 s | 56.9 s |
| 10 mm | Thin | 9.5 s | 20.5 s |
| 17 mm or more | Thick | 3.4 s | 5.6 s |
Thicker plate sheds heat faster and cools quickly, forming hard martensite prone to cracking. Preheat slows cooling, extending 3.4 s to 5.6 s for thick plate. Comparing FEM t8/5 with these values checks the heat source model and material constants.
Example 2: temperature change at which restrained steel yields
| Yield stress | Temperature change to yield |
|---|---|
| 250 MPa | 99 K |
| 355 MPa | 141 K |
Computed with Young's modulus 210 GPa and expansion coefficient 12×10⁻⁶/K. Welding changes temperature by more than 1,000 K, so the region near the weld always yields during heating and cooling. Residual stress magnitude is therefore set by the yield stress and its temperature dependence.
My thermo-elasto-plastic FEM gave longitudinal residual stress twice the yield stress. What is wrong?
Check the material model and how melted material is handled. Three causes are common. First, yield stress not reduced at high temperature: steel's yield drops sharply above 600 °C and is nearly zero near melting, so temperature-dependent data are essential. Second, not resetting the plastic strain history of material that melts and resolidifies: molten metal forgets prior deformation, so set plastic strain back to zero above the melting point (annealing). Third, the hardening law: heating and cooling reverse the loading direction, and isotropic hardening alone can accumulate hardening and overstate stress. Use kinematic or mixed hardening and check that post-cooling stress stays near the yield stress at that temperature.
Practical workflow
- Prepare temperature-dependent properties (conductivity, specific heat, Young's modulus, yield stress, expansion) up to near melting.
- Calibrate the moving heat source (e.g. Goldak's double ellipsoid) to the fusion-zone shape in a macrograph.
- Run the thermal analysis and compare t8/5 and peak temperatures with Rosenthal or measurements.
- Apply temperatures as loads in a thermo-elasto-plastic analysis with melt-zone history reset and a suitable hardening law.
- Compare with residual stress measured by X-ray diffraction or hole drilling, and with distortion.
“Multipass welds go one pass at a time”
In multipass welding of thick plate, each later pass tempers earlier ones and rewrites the residual stress distribution. Applying all passes as one heat source is fast but can give completely different surface and interior stress distributions. Adding deposit elements pass by pass and applying only that pass's heat is heavier but more likely to match measurements. With many passes, lumping neighbouring passes is a common simplification — use it while checking against measurements.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Room-temperature properties only | Stress overestimated | Data up to near melting |
| Keeping melt-zone history | Plastic strain accumulates | Reset history above melting |
| Uncalibrated heat source | Fusion zone and temperatures wrong | Match the macrograph |
| Isotropic hardening only | Stress inflates with cycling | Kinematic or mixed hardening |
| Wrong restraint | Distortion and stress swapped | Model fixtures as used |
I'd like to learn related topics.
Related pages include arc welding simulation, welding metallurgy simulation, spot welding simulation and thermal expansion and thermal stress.
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