Draw a straight line from the ridge purlin to the eave purlin and lower the first purlin below the ridge from it by one tenth of the rise; from that point draw a new line to the eave purlin and lower the next purlin by one twentieth; continue, halving each time.
drop₁ = H⁄10 · drop₂ = H⁄20 · drop₃ = H⁄40 · …
This is the zhé, the bending. Because each new line is drawn from the purlin just fixed, the depressions compound: the roof is steep near the ridge and flattens toward the eaves in a concave curve. Carpenters drew the construction full size on a wall or floor, the section of the roof appearing as a family of straight lines each slightly flatter than the last.
The result is a polygon of straight rafters, not a mathematical curve; the tiles and their mortar bed smooth it. The flattening at the eaves throws rainwater clear of the platform and lets low winter sun reach deep into the hall. These are two practical reasons often given for a form that is also, and above all, a matter of beauty.
For a great hall, measure the distance between the front and back eave purlins and raise the ridge purlin above them by one third of it; for lesser halls, by about a quarter.
rise = ⅓ × depth between eave purlins (halls); ≈ ¼ (+ a small addition) for ting-tang halls
Give each rafter span a fixed slope, increasing from the eaves to the ridge: half-pitch (5 jǔ) for the eave span, then about 6½ or 7, 7½, and 9 tenths for the span below the ridge.
rise of span k = run × (0.5, 0.65, 0.75, 0.9) for four spans
From the intermediate bracket set of the second bay the rafter ends advance progressively outward, so that at the corner the eave projects further (by 4 cun for a hall of one bay, 5 for three bays, 7 for five bays) and rises with the corner beam.
Give the hipped roof (wudian) to the highest halls, the hip-and-gable (xieshan) to the next, and the gabled roofs, overhanging (xuanshan) and flush (yingshan), to lesser buildings and houses; double the eaves for the very highest.
double wudian > double xieshan > wudian > xieshan > xuanshan > yingshan
Project the eaves from the centre of the eave purlin by 3½ chi for rafters of 3 cun diameter and 4 to 4½ chi for rafters of 5 cun; beyond them add flying rafters projecting six-tenths as far again.
eave projection ≈ 3½–4½ chi · flying rafters = 0.6 × eave projection
Raise a thatched roof by a third and a tiled roof by a quarter: the rougher and more absorbent the covering, the steeper the roof must be.
thatch: rise = ⅓ · tile: rise = ¼
Cite this page Chris Paton, “Depress each purlin: 1/10, 1/20, 1/40…”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/juzhe-depress. Text, drawings and models licensed under CC BY-NC 4.0.