What is the formula for curved surface correction for angle beam on pipe?

R = OD/2, r = R − depth, p = R sin θ; sound path SP = R cos θ − √(r² − p²); local angle at target φ = asin(p / r); central angle α = φ − θ; arc surface distance = R × α (α in radians); chord surface distance = 2 R sin(α/2); maximum angle reaching the bore: θ_max = asin(ID / OD). Exact sound path, central angle and arc surface distance for a shear beam scanned circumferentially from the outside of a pipe, plus the largest angle that still reaches the bore.

What should I check before relying on the result?

Applies to a probe on the OD firing circumferentially. Axial scanning on a cylinder sees a flat surface and needs no correction. The refracted angle used here is the angle at the entry point; contour the wedge to the pipe or the true entry angle will differ from the marking. Above θ_max the beam passes over the bore — the calculator then returns the path to the point of closest approach, not a real ID intersection. Verify against the governing edition: Circle geometry (constant perpendicular distance of a straight ray); scanning of curved surfaces per ISO 17640 Annex A and ASME BPVC Section V Article 4 T-434.1.7.2.

Curved surface correction for angle beam on pipe Calculator

Flat-plate trigonometry fails on pipe scanned in the circumferential direction. The beam leaves the outside surface at θ to the local radial normal and then travels in a straight line, but the normal rotates continuously around the circumference, so the angle between the beam and the wall keeps changing. The wall curves away from the beam and every depth is reached later, and further round, than a flat plate would suggest.

One geometric quantity makes the problem exact. A straight ray keeps a constant perpendicular distance from the pipe axis, p = R sin θ, so at any radius r the local angle to the radial direction obeys sin φ = R sin θ / r. From that follows the sound path SP = R cos θ − √(r² − R² sin²θ), and the angle subtended at the centre between the entry point and the target, α = asin(R sinθ / r) − θ. Multiplying α by R gives the true surface distance measured around the outside diameter.

The same relation sets a hard coverage limit. Since the ray never gets closer to the axis than p = R sin θ, it cannot reach the bore at all unless R sin θ ≤ r_i, that is θ ≤ asin(ID/OD). On a heavy-wall pipe a 70° or even a 60° probe simply passes over the root and out the far side without ever meeting the inside surface — the root is only covered by dropping the angle or by working from the second leg with a lower angle probe.

Two practical points follow. The wedge must be contoured to the pipe or the entry angle is not what is marked, and the surface distance you chalk on the pipe is an arc, not a chord — the gap between the two grows with both diameter and central angle, from a few tenths of a millimetre on small pipe to several millimetres on large-diameter work, and the difference between either and the flat-plate answer can be tens of millimetres.

Shear wave

Worked example

Outside diameter	200 mm
Wall thickness	20 mm
Refracted shear angle at the OD surface	45 deg
Target depth below the OD surface	20 mm
Closest approach to axis	70.71 mm
Sound path to target (curved)	33.29 mm
Sound path if the surface were flat	28.28 mm
Curvature error in sound path	5.01 mm
Beam angle to the wall at the target	62.11 deg
Central angle entry to target	17.11 deg
Surface distance around the OD (arc)	29.87 mm
Surface distance straight across (chord)	29.76 mm
Surface distance if the surface were flat	20 mm
Maximum angle that still reaches the bore	53.13 deg

200 mm OD, 20 mm wall, 45° probe aimed at the bore. R = 100, r = 80, p = 100 sin45° = 70.71 mm. Sound path = 100 cos45° − √(80² − 70.71²) = 70.711 − √1400 = 70.711 − 37.417 = 33.29 mm, against 20/cos45° = 28.28 mm on flat plate — 5.01 mm longer. The beam meets the bore at asin(70.71/80) = 62.11°, so the central angle is 62.11 − 45 = 17.11° and the arc distance is 100 × 0.29871 rad = 29.87 mm (chord 29.76 mm) instead of the flat-plate 20.00 mm. Any angle above asin(160/200) = 53.13° would miss the bore altogether.

Use at your own risk — verify before you act

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