In a corbelled earthbag dome each course steps inward from the one below, and with a step limit of one third of the bag width the geometry decides how far up the dome can be built. For a 3 m hemispherical dome built from 40 cm wide bags in 11 cm courses, the step limit is passed about 1.16 m up, so the top cannot be closed by corbelling alone and needs a pointed profile or a separate cap over the remaining opening.
Part of Earthbag Building
A corbelled dome closes by shifting each bag course slightly inward; the arithmetic of how far each course must move follows from the dome’s profile and the course height.
The Parent Rule Is A Step Limit, And Geometry Decides Where It Binds
The Earthbag Building overview gives a maximum corbel of one third of the bag width per course, and says more than that risks collapse before the dome closes. That is a rule of thumb, not a tested threshold, and this article does not validate it. What it does is test a dome profile against the rule, which is arithmetic with no further assumption.
For a profile with inner radius r(z) at height z, each course of height h moves in by the difference in radius between its bottom and its top. For a hemisphere of radius R, the surface at height z leans from vertical by an angle with sine z/R, and the horizontal step per course is approximately h x tan(lean) = h x z / r.
What You Need
- The dome’s inner radius R.
- The tamped course height h: the parent article gives 10-12 cm.
- The bag width w: the parent article gives 40 cm walls.
- A calculator or a table of square roots, and the habit of tabulating before building.
Method: Tabulate The Steps Before You Build
Step 1: Choose the profile (hemisphere, pointed or other) and the inner radius.
Step 2: Compute the inner radius at the top of each course: r(k) = square root of (R2 minus z2), where z = k x h for a hemisphere.
Step 3: Take the step for each course as r(k-1) minus r(k).
Step 4: Compare each step with the limit, w / 3.
Step 5: Find the course where the limit is passed, and decide how the top will be closed.
Worked Example
Inputs (assumed): hemisphere, R = 1.5 m (3 m diameter), h = 0.11 m, or 11 cm (the middle of 10-12 cm), w = 0.40 m.
- Limit: w / 3 = 0.40 / 3 = 0.133 m per course.
- The limit corresponds to a lean from vertical where tan(lean) = 0.133 / 0.11 = 1.21, so the lean is 50.5 degrees.
- For a sphere, sin(lean) = z / R, so z = 1.5 x sin(50.5 degrees) = 1.5 x 0.771 = 1.157 m, about 1.16 m.
- Radius left open at that height: r = 1.5 x cos(50.5 degrees) = 1.5 x 0.636 = 0.955 m, a diameter of about 1.9 m.
Step per course for this dome:
| Course | Height of top (m) | Inner radius (m) | Step (m) |
|---|---|---|---|
| 4 | 0.44 | 1.434 | 0.029 |
| 8 | 0.88 | 1.215 | 0.073 |
| 10 | 1.10 | 1.020 | 0.107 |
| 11 | 1.21 | 0.887 | 0.133 |
| 12 | 1.32 | 0.712 | 0.174 |
| 13 | 1.43 | 0.453 | 0.260 |
The first courses step 1.5-3 cm, which matches the parent article’s typical figure for the compass method. The step reaches the limit at course 11, and courses 12 and 13 would have to step 17 and 26 cm, beyond it. A smooth hemisphere has a horizontal tangent at its apex, so any smooth round dome meets this problem at its crown; only a pointed profile avoids it.
A straight cone at exactly the limit slope is the benchmark. It rises 0.11 m for every 0.133 m inward, so over a 1.5 m radius it rises 1.5 x (0.11 / 0.133) = 1.24 m, in about 1.5 / 0.133 = 11 courses. A pointed dome whose every part is steeper than this cone passes the test everywhere.
Shallow Profiles Make The Test Harder
The parent article suggests a shallow profile (height less than half the diameter) for domes of 4 m and larger. For corbelling the geometry points the other way. Take a spherical cap of radius a = 1.5 m and rise H = 0.9 m. Its sphere radius is (a2 + H2) / (2H) = (2.25 + 0.81) / 1.8 = 1.7 m. The same 50.5 degree lean limit is passed at z = 1.7 x 0.771 - (1.7 - 0.9) = 0.51 m, which is 57% of the rise, against 77% of the radius for the hemisphere. Flattening the profile brings the limit sooner, so the shallow-profile advice must be judged on structural thrust grounds, and the step test applied on top of it.
Common Mistakes
- Closing a hemisphere by corbelling alone. The last courses overhang beyond the limit and may fall during construction.
- Treating the compass cord as the whole answer. The cord only reproduces the profile you chose; it does not check the step.
- Using one step for every course. The step grows from a few millimetres at the base to the limit near the top.
Safety
Corbelled courses can collapse before the dome closes
A partly built dome has no ring action until it closes, so each overhanging course rests only on friction and the barbed wire. Do not work under the unfinished crown, prop or brace upper courses, and treat any course where the step approaches the limit as unsafe without a form beneath it.
What’s Next
- Learn next: Compass and Form Guides — setting out the profile you tabulated
- Combine with: Dome Building — the same problem in masonry