Masonry arch
A segmental, semicircular or jack (flat) arch of unreinforced masonry over an opening under a uniform load: the line of thrust, the compression in the ring, the skewback joint, and the masonry pier beside the opening or a steel tie that takes the horizontal thrust. US and metric units.
Edition and method. The BIA Technical Notes on arches (TN 31 series) are not on hand: the thrust-line method is written from knowledge and marked "to confirm against the code text"; the stresses use TMS 402 Chapter 8 (unreinforced, allowable stress) through the shared masonry library (402-13 text with 402-22 numbering where confirmed). The method switch is hidden: allowable stress only, load combinations D and D + L.
Inputs of the arch #
- Arch: shape; span between springings; the rise of the axis (segmental; a semicircle rises half the span, a jack arch is flat); the ring's radial depth; the arch width (the wall thickness); for a jack arch the skewback angle from vertical; f′m; mortar (for the abutment's allowable tension).
- Loads: dead and live load per foot, uniform over the span. Whether the arch carries the full height of wall above or only the arching triangle is your judgment: enter the load that results, with the arch's own weight.
- Abutment: a steel tie (diameter, Fy) or the masonry beside the opening: its length to the wall end, the next opening or a joint (the end pier governs), the springing's height above the pier's base, the masonry above the springing over it, and the wall weight.
Auto-design varies the ring depth and the tie diameter. Objective: the shallowest ring, then the smallest tie. The span, rise and shape are the architect's and are kept.
Checks performed #
| Check | Basis |
|---|---|
| Thrust line within the middle third | The least-thrust parabola through points within the middle third at the crown and the springings (a 21 × 21 grid), tested at 41 sections; BIA TN 31 series (to confirm) |
| Ring compression | fa/Fa + fb/Fb ≤ 1, Fa = 0.25 f′m, Fb = 0.45 f′m, the resultant along the thrust line at its offset (TMS 402 8.2.4) |
| Skewback joint shear | 1.5 V / A ≤ min(1.5 √f′m, 120 psi, 37 psi + 0.45 Nv/An) (8.2.6.2) |
| Steel tie | AISC 360-22 D2, yielding of the gross area |
| Abutment pier | Net tension (Table 8.2.4.2, solid units), compression, and sliding at the springing bed joint |
H = w S² / (8 f) with f the rise of the thrust line; for a jack arch f = d/3 and H = 3 w S² / (8 d). The drawing is the arch elevation with the line of thrust.
Limitations #
- Uniform load only: concentrated loads and live load on half the span (which moves the thrust line) are not covered.
- The thrust line's offset is measured vertically and converted by the axis slope; near the springings of a semicircular arch, where the axis is steep, this approximation is coarse.
- Unreinforced arches only; the wall above acting as a deep beam, out-of-plane loads, rings acting separately, Gothic and elliptical shapes, and settlement of the abutments are not covered. The tie's end anchorage is not designed.
Verification examples #
Verification status: 10 values compared in 3 examples (0 published, 3 hand calculations from the cited equations): 10 within tolerance, 0 recorded deviations. No published example: every value is a hand calculation. That checks the program against this reading of the code, not the reading itself. Code basis: not yet recorded for this module.
No published arch example is on hand: hand calculations of the jack arch thrust (3 w S² / (8 d)), a shallow segmental arch's least thrust (f + d/6 + d/(6 cos θs)), the ring and skewback stresses, the abutment pier, and the tie.