Measuring one log: mid-diameter, Huber and Smalian

The usual way to find the volume of a single log in Central Europe is to treat it as a cylinder with the diameter taken at the middle. This lesson follows the German framework agreement (RVR, Germany, Europe, 5th edition 2023), compares it with the end-area method of Smalian, and works through the rounding and allowance rules that change the billed volume.

The Huber formula

The Huber-type formula multiplies the cross-section area at the middle of the log by its length: V = (pi/4) x d^2 x L. With d in centimetres and L in metres this is V (m3) = 0.7854 x d^2 x L / 10,000. Under the RVR the result is solid volume without bark (Fm o.R.).

Worked example. Mid-diameter 30 cm, length 5.00 m: 0.7854 x 900 x 5 / 10,000 = 0.3534 m3, reported as 0.35 m3.

The method needs one measurement point and is quick. For a log with bulges, sweep or butt flare it is an approximation, which is why mills with scanners add up thin slices (lesson 3).

How the RVR reads the diameter

The RVR measures the mid-diameter without bark at the middle of the log. The practical rules:

• One horizontal caliper reading up to 19 cm and two perpendicular readings from 20 cm.
• Round down to whole centimetres. A reading of 36.8 cm is billed as 36 cm.
• Length starts at the middle of the Fallkerb (the felling notch).
• Length allowance: 1% of order length, rounded up to whole centimetres, minimum 10 cm, excluded from volume.

The allowance exists because a saw cut in the forest is never exact.

Worked example. A spruce log has order length 4.50 m. One percent is 4.5 cm, below the minimum, so the allowance is 10 cm and the delivered log may be 4.60 m. The mid-diameter reads 36.8 cm. Correct billing uses d = 36 cm and L = 4.50 m: 0.7854 x 1,296 x 4.5 / 10,000 = 0.458 m3. A mill that wrongly used 36.8 cm and 4.60 m would get 0.7854 x 1,354.24 x 4.6 / 10,000 = 0.489 m3, which is 6.8% too much (own calculation). Two small conventions together account for almost 7%.

Diameter classes

The RVR groups logs by mid-diameter: D0 (up to 9 cm), D1a (10-14), D1b (15-19), D2a (20-24), D2b (25-29), D3a (30-34), D3b (35-39), D4 (40-49), D5 (50-59), D6 (60-69), D7 (70-79) and D8 (80 cm and above). The 36 cm log above is D3b. Price lists are written per class, so rounding down also decides which price applies: 39.9 cm is billed as 39 and stays in D3b, while a reading of 40 moves the log to D4.

Smalian: two end areas

Smalian's formula averages the end areas: V = (A_butt + A_top) / 2 x L, with A = (pi/4) x d^2. The portal's yield article only reports that Kilborn (2002, Alaska, North America) measured log volume by Smalian's formula; the formula itself is textbook mathematics, not printed in the portal's articles.

Worked example. Butt 34 cm, top 26 cm, L = 5 m. A_butt = 0.09079 m2 and A_top = 0.05309 m2, so V = (0.09079 + 0.05309) / 2 x 5 = 0.3597 m3. Huber with the 30 cm mid-diameter gives 0.3534 m3, so Smalian is 1.8% higher on this straight-taper log (own calculation).

The lesson is practical: a yield figure or a price is only comparable if the volume formula is the same. 60% yield on Smalian volumes and 60% on Huber volumes are not the same quantity.

Sampling methods for short wood

For short wood the FVA leaflet 49/1997 (Germany, on waldwissen.net) describes the end-face method, the mid-diameter sample and the section stacked-volume method; all three can serve as sales, billing or control measures if written into the contract.

The RVR is stricter: it restricts sampling for billing to exceptional cases (calamities, lost original data, disproportionate effort), by mutual agreement, and recognises only the end-face and mid-diameter methods. The "Heilbronner Sortierung" could not be verified in an open source and is not described here.

Key points

• Huber volume is 0.7854 x d^2 x L / 10,000 (d in cm, L in m); a 30 cm, 5.00 m log is 0.35 m3.
• The RVR rounds mid-diameter down to whole centimetres and excludes the length allowance (1% of order length, minimum 10 cm) from volume.
• Ignoring both overstated a 4.50 m, 36.8 cm log by 6.8% (own calculation).
• Smalian (end areas) was 1.8% above Huber for the 34/26 cm, 5 m example; never mix volumes from different formulas.
• The RVR allows sampling for billing only exceptionally.