Arch analysis
Three-hinged, two-hinged and fixed (hingeless) arches of parabolic or circular shape, with the supports at the same level, under vertical and horizontal point loads, uniform loads along the span and the arch's own weight: the reactions, the horizontal thrust, and the moment, normal thrust and radial shear along the arch. Use it for steel, glulam and concrete arches, arched roof ribs and frames, and to see how far a load pattern takes the arch from its thrust line. The forces are for the design of the arch in its material module.
Inputs of the arch analysis #
- Arch: three-hinged (supports and a hinge at the crown or at a point you give), two-hinged (pinned supports) or fixed.
- Shape, span L and rise f: parabolic, y = 4 f x (L − x) / L², or a circular segment through the supports and the crown (at most a half circle). The geometry is the arch's axis.
- Section (two-hinged and fixed): E, A and I, constant or growing as sec θ from the crown (I = I0 sec θ, the classical assumption behind the textbook formulas). Axial shortening can be neglected, as in those formulas; with it, a shallow arch carries less thrust. A three-hinged arch is statically determinate and needs no section.
- Loads: uniform loads per horizontal length over the span or part of it, point loads (P downward, H to the right) and the weight of the arch along its length. One load set, as entered: run the arch once per load case or combination.
The arch analysis has no auto-design: there is nothing to size.
Checks performed #
| Result | Basis |
|---|---|
| Reactions, horizontal thrust (end moments of a fixed arch), the moment at the hinge, equilibrium | Statics; redundants from a chord model of the arch (the plane-frame library, first order) |
| Moment, normal thrust and radial shear at every station, both sides of each point load | Statics of the part left of the section on the true curve: M = M0 − H y (+ end moment), N = Fx cos φ + Fy sin φ, V = Fy cos φ − Fx sin φ (Udoeyo 2020, Eqs. 6.3 to 6.5) |
| Largest positive and negative moment, thrust, shear | Over the stations |
Moment is positive with tension inside the arch (on the intrados); the normal thrust is positive in compression.
Limitations of the arch analysis #
- Supports at the same level and rigid; no ties, elastic supports, settlement or temperature.
- First-order elastic analysis: no second-order effects, in-plane or out-of-plane buckling of the arch. Check the arch's stability in its member design.
- The redundants of two-hinged and fixed arches come from 64 chords plus a node at every load (their error falls as the square of the chord length); the three-hinged arch is exact.
- No member strength is checked.
Verification examples #
Verification status: 23 values compared in 9 examples (3 published, 6 hand calculations from the cited equations): 23 within tolerance, 0 recorded deviations. Code basis: the provisions it uses were read in the target edition.
| Example | Source | Values |
|---|---|---|
| Three-hinged parabola, point and partial load (metric) | Udoeyo, Structural Analysis (2020), Example 6.2 | Ay, By, thrust, normal thrust and radial shear left of the load |
| Three-hinged parabola, uniform load | Udoeyo Example 6.3 | Ay, thrust, moment at 18 ft (zero; the book's −75 lb-ft is rounding) |
| Three-hinged parabola, two point loads | Udoeyo Example 6.4 | Ey, Ay, thrust, moments under the loads |
| Two-hinged parabola, I sec θ | Hand calculation (H = ∫M0 y dx / ∫y² dx) | wL²/8f, wL²/16f, 25PL/128f and the crown moment |
| Fixed parabola, I sec θ | Hand calculation (elastic equations) | 15PL/64f, end moment PL/32, crown moment 3PL/64; wL²/8f |
| Three-hinged circular arch | Hand calculation | Thrust, moment at the quarter point |
The two-hinged and fixed closed forms are derived from the compatibility equations; a textbook that prints them (Timoshenko and Young, Theory of Structures, or a structural analysis text) is not on hand.