A stone arch stands as long as a line of thrust, the path along which the compressive force travels, can be drawn entirely inside the stone ring; it fails by turning into a mechanism when 4 hinges form, almost never by crushing. The horizontal thrust of a parabolic arch is H = wL² / 8f, so a 6 m span with a 3 m rise under 25 kN/m pushes outward with 37.5 kN per metre of width, and halving the rise doubles it. Check an arch by its geometry and its abutments, not by dividing a crushing strength by 3.
Part of Bridge Types & Construction
The statics of masonry arches: thrust, the line of thrust, minimum ring thickness and the abutment forces that keep a stone ring standing.
Stone Arches Fail By Changing Shape, Not By Crushing
Stone is strong in compression and has almost no useful tension. An arch arranges the stones so that every joint is squeezed. The parent article states this: each stone pushes on its neighbours, the forces flow into the abutments, and the abutments must resist the outward thrust.
The modern treatment, due to Jacques Heyman, assumes three things about masonry: it has no tensile strength, its compressive strength is so high that crushing is not the limit, and stones do not slide on each other. Under those assumptions, an arch is stable if you can draw some line of thrust through it, in equilibrium with the loads, that stays entirely within the thickness of the ring. This is the “safe theorem” of Heyman’s papers “The stone skeleton” (1966) and “The safety of masonry arches” (1969). I am citing both from memory and have not rechecked the papers here. An arch collapses when no such line exists, because the line touches the edge of the ring at 4 places, hinges open there, and the ring becomes a mechanism.
The Hooke principle makes the line visible. A flexible chain hung from two points takes a shape in pure tension. Invert it and you get a shape in pure compression. Robert Hooke published this idea in 1675 as an anagram, “as hangs the flexible line, so but inverted will stand the rigid arch”.
What You Need
- A string or light chain and small weights, for the hanging model.
- A scale drawing of the arch ring, abutments and fill.
- The weight of each part — see Common Material Weights for stone and fill.
- The parent’s 3 for stone arches — treated below as a crushing check, which is the wrong test for geometry.
Method: Find The Line Of Thrust With A Hanging Chain
Step 1: Scale the arch Draw half the span and rise at a convenient scale, such as 1:20, on a board. Mark the ring thickness above and below the centre line.
Step 2: Divide the ring into blocks and weigh them Cut the ring and the fill above it into vertical slices. Compute the weight of each slice, including fill and expected traffic.
Step 3: Hang the weights Hang a light cord across the board at the spring points. Attach a small weight, proportional to each slice, at each slice’s centre line. Adjust the sag until the cord passes through the crown and springing points you choose.
Step 4: Read the inverted curve Trace the cord and mirror it. If the mirrored curve lies inside the ring everywhere, the arch is stable under those loads. If it leaves the ring, add thickness, add weight over the haunches, or change the rise.
Step 5: Take the end force The cord tension at the springing is the force the abutment must resist; its horizontal part is the thrust.
Worked Example: A 6 m Arch At 3 m Rise
Inputs, mine: span L = 6 m, rise f = 3 m (semicircle, radius 3 m); ring depth 0.4 m, which is 1/15 of the span, the lower end of the range in Arch Construction; solid stone 2,600 kg/m³, inside the granite and limestone ranges of Material Properties; a uniform load of w = 25 kN per metre of span per metre of width, made of the ring (17.1 kN/m: 4.02 m³ of ring per metre of width × 2,600 × 9.81 / 6 m) and about 8 kN/m for fill, surfacing and traffic, an assumption of about 0.4 m of fill at 20 kN/m³. The parabolic formula is exact for a parabola under load spread evenly along the span and only an estimate for a semicircle, so treat the thrust as an order-of-magnitude check.
- Horizontal thrust: H = w L² / (8 f) = 25 × 6² / (8 × 3) = 25 × 36 / 24 = 37.5 kN per metre of width.
- Vertical reaction: V = w L / 2 = 25 × 6 / 2 = 75 kN per metre of width.
- Resultant at the springing: √(37.5² + 75²) = 83.9 kN per metre, at arctan(75 / 37.5) = 63.4 degrees above horizontal.
- Rise one-quarter of the span, as Arch Bridge recommends as a minimum: f = 1.5 m, so H = 25 × 36 / (8 × 1.5) = 75 kN per metre, double. Flatter arches push harder.
Sliding of the abutment. The abutment base carries V plus its own weight W. The friction needed is μ = H / (V + W). With W = 100 kN per metre: μ = 37.5 / 175 = 0.21; with a friction coefficient of 0.5 (an assumption) the margin is 0.5 / 0.21 = 2.3. With W = 50 kN per metre: μ = 0.30 and the margin is 1.7.
Crushing. The average stress at the springing is 83.9 kN / 0.4 m = 210 kPa, or 0.21 MPa. If the line of thrust is pressed to within 10% of the ring depth from the edge, only 3 × 0.04 m = 0.12 m of the joint carries load. The edge stress is then 2 × 83.9 / 0.12 = 1,400 kPa, or 1.4 MPa. Material Properties gives limestone as 5,000-15,000 PSI (about 34-103 MPa) and granite as 15,000-30,000 PSI (about 103-207 MPa). The stress is below 5% of those values, so a factor of 3 on crushing is not the margin that matters.
Minimum thickness. Heyman’s result for a semicircular ring of uniform thickness carrying only its own weight is a minimum ratio of thickness to radius of about 0.1075 (recalled, and quoted as 0.106-0.11 in some texts). For R = 3 m that is 0.32 m. A 0.4 m ring gives a ratio of 0.4 / 0.32 = 1.24, and a 0.5 m ring (1/12 of the span) gives 1.55. Those are geometric factors of safety, not load factors, and they are the right kind of number for an arch. Fill above the haunches generally improves stability, but a heavy single load near a quarter point can reduce it, which is why the chain model with the real loads is worth building.
Common Mistakes
- Applying the parent’s 3 as a crushing factor — the factor is real but it is the wrong place to look; 3 on a stress that is already below 5% of strength changes nothing, while geometry and abutments govern.
- Flattening the arch to save stone — halving the rise doubles the thrust.
- Ignoring abutment movement — if the abutments spread even a little, hinges open and the ring may become a mechanism.
- Using a parabola formula as a design — it is an estimate. Build the chain model.
- Heavy loads at the quarter point — a cart or a stack of stone placed there can open a hinge.
Safety
Arches collapse as mechanisms
A masonry arch can look sound and then fail suddenly when an abutment moves or a hinge opens. Cracks that open on the underside at the crown and on the top surface at the haunches are typically the signature of hinges forming. Keep people off and out from under an arch that shows them, and do not load it until the cause is found.
What’s Next
- Learn next: Centering & Formwork — holding the ring in position while it is built
- Combine with: Abutment Construction — resisting the thrust computed here