Welding Distortion and Residual Stress Analysis
Theory: weld-zone shrinkage leaves a shrinkage force
Overview
Does predicting welding distortion always need a large thermo-elastic-plastic analysis?
If you need the final shape in detail, yes; but there is a simple way to get the order of magnitude first. Metal near the weld gets hot, tries to expand, is held back and yields in compression. On cooling it wants to shorten and leaves a pull on its surroundings — the shrinkage force, treatable as a single force along the weld line. Knowing it, plate shortening and camber follow from beam formulas. Plastic-zone width comes from how far the welding heat raises temperature. This method helps identify troublesome joints before weld sequence and fixtures are studied.
Peak temperature and plastic zone
Thin-plate peak-temperature formula (Adams). q/v is net heat input (J/mm), ρc volumetric heat capacity (4.71×10⁻³ J/mm³K for steel), t thickness, y distance from the weld line and T_m − T₀ the rise to melting (1,480 K). The plastic zone is where peak temperature rise exceeds ΔT_cr (119 K for 300 MPa yield, 210 GPa modulus and 12×10⁻⁶/K expansion).
Shrinkage force, shortening and camber
y_p is plastic half-width, B plate width, L length and e offset of the weld from the plate centre. Plastic width is proportional to heat input, so shrinkage force is proportional to heat input and independent of thickness.
The ship that split at the pier
During the Second World War the United States built ships in great numbers by welding instead of riveting. In January 1943 the newly completed tanker Schenectady split in two amidships with a loud report while moored at a calm pier. Low temperature making the steel brittle, stress-concentrating details and welding residual stress are thought to have combined. The accident drove major advances in research on low-temperature toughness of steel and welding residual stress.
Worked examples
Example 1: heat input and distortion (10 mm thick, 2 m long, 300 mm wide)
| Net heat input | Plastic half-width | Shrinkage force | Shortening | Camber (100 mm offset) |
|---|---|---|---|---|
| 400 J/mm | 15.9 mm | 95 kN | 0.30 mm | 1.01 mm |
| 800 J/mm | 31.7 mm | 190 kN | 0.60 mm | 2.02 mm |
| 1,200 J/mm | 47.6 mm | 286 kN | 0.91 mm | 3.02 mm |
800 J/mm is a 25 V, 200 A, 5 mm/s arc (1,000 J/mm) times 0.8 efficiency. Shrinkage force works out to about 0.24 kN per J/mm. With the weld on the centre line camber is zero and only the 0.60 mm shortening remains.
Example 2: thickness reduced from 10 to 6 mm (800 J/mm)
| Thickness | Plastic half-width | Shrinkage force | Shortening |
|---|---|---|---|
| 10 mm | 31.7 mm | 190 kN | 0.60 mm |
| 6 mm | 52.9 mm | 190 kN | 1.01 mm |
At 6 mm the plastic zone reaches the plate edge (50 mm from the weld), so this simple formula cannot estimate camber reliably.
Thin plates need less heat, yet distort more?
At equal heat input, yes. In thin plate the same heat concentrates in less volume, so the heated region widens and the plastic zone grows inversely with thickness. Shrinkage force (plastic width × thickness × yield stress) thus stays the same regardless of thickness, while the plate's resistance to shortening and bending is proportional to thickness, so thin plates distort more. Thin plates may also buckle where shrinkage force acts in compression, producing wavy distortion. That's why lowering heat input (e.g. laser or narrow beads) is particularly effective on thin plate. In practice weld sequence, restraint, tack welds and multi-pass welding change the outcome, so the final shape and residual-stress distribution need a thermo-elastic-plastic analysis that follows the temperature history. It is efficient to estimate with the simple formulas first, then analyse troublesome joints in detail.
Modelling workflow
- Use simple shrinkage-force formulas to find joints likely to distort most.
- Solve temperature history with a moving weld heat source.
- Run thermo-elastic-plastic analysis with temperature-dependent yield and modulus.
- Include weld sequence, restraint and tacks to obtain distortion and residual stress.
- Compare with measured trial distortion and tune heat-source efficiency and shape.
“Camber halved just by changing the order”
On a long box beam, finished camber far exceeded tolerance. All top-flange welds were completed before the bottom ones, so the top shrinkage force alone bowed the beam first, and it set in that shape. Comparing sequences with shrinkage-force estimates and thermo-elastic-plastic analysis, switching to alternating top and bottom welds in short steps roughly halved the camber — distortion reduced by procedure alone, without changing heat input.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Using arc heat input directly | Distortion overestimated | Apply efficiency |
| Room-temperature constant properties | Residual stress wrong | Temperature-dependent properties |
| Ignoring weld sequence | Camber wrong | Solve in sequence |
| Missing thin-plate buckling | Wavy distortion not predicted | Add buckling analysis |
| Simple formula near edges | Plastic zone reaches edge | Switch to detailed analysis |
I'd like to learn related topics.
Related pages include thermal residual stress, additive manufacturing residual stress, thermal buckling, glass tempering and thermal assembly.