Welding math
Weld Reinforcement and Convexity
The crown on a weld face is close enough to a parabolic cap that two thirds of width times height gives its area, which turns an eyeballed hump into pounds of filler and a measurable convexity ratio.
The formula
A_reinf = (2/3) x w x h, convexity% = 100 x h / wThe extra metal in a weld crown approximates a parabolic cap of area two thirds the face width times the crown height, and convexity is the crown height expressed as a percentage of the face width.
Work it out
How much metal is in the crown, and how proud it stands.
- Convexity
- 12.0%
- Reinforcement area
- 0.0200 in2
The same arithmetic as the worked examples above, on your own numbers. It reports geometry and physics, never an acceptance limit: what a joint is allowed to be comes from the job's own code and contract documents.
What each symbol means
| Symbol | Meaning | Units |
|---|---|---|
| A_reinf | Cross-sectional area of metal above the base line | in^2 |
| w | Width of the weld face across its base | in |
| h | Height the crown stands above that base line | in |
| A_total | Groove or fillet area plus the crown | in^2 |
Why it works
A weld crown is not a triangle and not a semicircle. Surface tension pulls it into a smooth arch that a parabola models well, and the area under a parabolic cap is exactly two thirds of the enclosing rectangle. So a crown a sixteenth of an inch tall on a three-quarter-inch face carries two thirds of that product, and that figure adds directly onto the groove area beneath it.
Convexity as a ratio of height to width is the number that matters for fatigue and for stress concentration. A tall narrow crown makes a sharp re-entrant angle where the weld toe meets the plate, and sharp toe angles concentrate stress. A low wide crown blends in. Reporting the ratio rather than the raw height is what lets a five-thousandths crown on a narrow bead be recognised as worse than a ten-thousandths crown on a wide one.
Reinforcement is one of the few weld features that costs money and often subtracts value. It buys no design strength, because groove welds are sized on the thickness of the joined members, and it hurts fatigue life by steepening the toe. What acceptance limits apply to a given job come from that job's contract documents; the geometry here just tells you what you are looking at.
Worked examples
What you are given, what you do with it, and what the answer actually tells you. Cover the steps and try it before you read them.
Example 1: Crown on a groove weld face
Given
- Face width
- 0.75 in
- Crown height
- 0.0625 in
- Groove area beneath
- 0.125 in^2
Working
- Approximate the crown as a parabolic cap: (2/3) x 0.75 x 0.0625 = 0.03125 in^2.
- Add it to the groove area beneath: 0.125 + 0.03125 = 0.15625 in^2.
- Express convexity as a ratio: 100 x 0.0625 / 0.75.
Result
0.03125 in^2 of reinforcement, 8.33 percent convexity, 0.15625 in^2 total
The crown adds a quarter again on top of the groove metal and contributes nothing to the design strength. A weld that looks only slightly proud is carrying a surprising share of the total deposit.
Example 2: Convex fillet face
Given
- Face width
- 0.4 in
- Crown height
- 0.05 in
Working
- Cap area: (2/3) x 0.4 x 0.05 = 0.01333 in^2.
- Convexity ratio: 100 x 0.05 / 0.4 = 12.5 percent.
Result
0.01333 in^2 of extra metal at 12.5 percent convexity
Twelve and a half percent is a distinctly humped fillet. The metal is wasted, and worse, the steep toe angle it creates is exactly where a fatigue crack would start.
In practice
- Measure crown height with a gauge that bridges both plate surfaces, not from the groove edge, or a slightly mismatched joint will read as reinforcement.
- Convexity and reinforcement are geometry; whether a particular amount is acceptable is a contract question answered by the job's own documents.
- Grinding reinforcement flush is common on fatigue-loaded joints, but grinding into the base metal turns a cosmetic fix into a thickness deficiency.
- A consistently proud cap across a shop usually points at technique or parameters rather than at individual welders: too slow a travel, or too much wire for the voltage.