Apply a uniformly distributed load and an optional crown point load to a parabolic three-hinged arch and study the vertical reaction and horizontal thrust at its supports. Adjust the span, rise and loads to see in real time how an arch carries load in compression rather than bending, and how hard it pushes its supports outward.
Parameters
Span L
m
Horizontal distance between the supports
Rise (arch height) h
m
Height from the supports to the crown
Distributed load w
kN/m
Load distributed over the full span
Crown point load P
kN
Point load applied at the crown hinge
Results
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Vertical reaction (kN)
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Horizontal thrust H (kN)
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Resultant reaction (kN)
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Reaction angle (deg)
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Rise-to-span ratio h/L
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Arch shape verdict
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Three-hinged arch — loads and reactions
Parabolic three-hinged arch. It shows the three hinges (two at the supports and one at the crown), the distributed load over the full span, the crown point load, and the vertical reaction and horizontal thrust at each support.
Vertical support reaction V and horizontal thrust H. w: distributed load, L: span, h: rise, P: crown point load. The vertical reaction depends only on the load, while the horizontal thrust is inversely proportional to the rise.
Resultant support reaction R and its angle θ from the horizontal. The three hinges — a pin at each support and one at the crown — make the arch statically determinate.
When a parabolic arch carries a uniformly distributed load over the full span, its axis coincides with the funicular (the natural thrust line) of that load, and the bending moment is zero everywhere — the arch is in pure compression, the ideal state.
What is the Three-Hinged Arch Simulator?
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Why are the "arches" you see in stone bridges and tunnels so curved? Wouldn't a straight beam do the job?
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Good question. A straight beam carries load by "bending". In bending, the top half of the section shortens and the bottom half stretches — only part of the material is really working, so it is quite an inefficient way to carry load. An arch, by curving, creates a path for the load so that it is carried almost entirely by "axial compression". Stone and concrete are extremely strong in compression, so an arch deliberately plays to their strength.
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So carrying load in compression is a big win. Then surely an arch is all upside?
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There is a price for that efficiency. An arch generates a "horizontal thrust". When loaded, it tries to push its supports outward at an angle. Look at the "Horizontal thrust H" in the tool on the left — by default it pushes 375 kN sideways. Unless the abutments, buttresses, or a tie connecting the two supports are strong enough to resist this thrust, the arch spreads outward and collapses.
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I see. So what is the "three-hinged" part? A hinge like a door hinge?
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Yes — a joint that allows rotation. A three-hinged arch has one hinge at each support and one more at the crown, three in total. That is very clever: adding three hinges makes the structure "statically determinate". All the reactions can be found from the equilibrium equations alone, with no knowledge of how stiff the arch is. Notice that the calculation in the tool below uses neither a Young's modulus nor a cross-section.
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What is the benefit of not needing the stiffness?
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It matters a lot in practice. In a fixed arch, even a slight support settlement, or expansion and contraction with summer and winter temperatures, builds up large "secondary stresses" inside the structure. A three-hinged arch simply lets its hinges rotate a little to absorb those movements, so almost no secondary stress develops. That is why it is favoured for bridges on soft ground or in big temperature swings, and for gymnasium roofs. There is also a beautiful result: when a parabolic arch carries a uniform load over the whole span, the bending moment is zero everywhere — pure compression. Raise the rise and the thrust falls; a low, flat arch pushes its supports harder.
Frequently Asked Questions
For a parabolic three-hinged arch carrying a uniformly distributed load w over the full span, taking moments about the crown hinge gives the horizontal thrust H = wL²/(8h), where L is the span and h is the rise. A crown point load P adds a term PL/(4h). Thrust is inversely proportional to the rise h, so a taller arch pushes its supports outward less than a low, flat one.
A three-hinged arch has a pin at each support and a third hinge at the crown — three hinges in total. There are four unknown reactions (a vertical and a horizontal component at each support), but in addition to the three global equilibrium equations there is one more condition: the bending moment at the crown hinge is zero. Four equations then solve four unknowns, so the reactions are found without using the material stiffness at all. This is why a three-hinged arch develops almost no secondary stress under support settlement or temperature change.
The vertical support reaction depends only on the load, so it does not change with the rise. The horizontal thrust H = wL²/(8h) is inversely proportional to the rise h — double the rise and the thrust is halved. A tall, steep arch pushes its supports (abutments or buttresses) outward less, but the arch itself becomes longer, so buckling and construction become harder. The rise-to-span ratio in this tool indicates whether the arch is flat, standard or tall.
The funicular — the natural thrust line — of a uniformly distributed load over the full span is exactly a parabola. If the axis of the arch coincides with this funicular, the load is carried at every section by axial compression alone and the bending moment is zero everywhere. This means the arch is in pure compression, the ideal structural state. When the load distribution changes, the thrust line changes too, and a bending moment appears wherever the arch axis departs from it.
Real-World Applications
Bridges: Three-hinged arches have long been used in steel and concrete arch bridges. Because the hinges at the two supports and the crown absorb support settlement and thermal expansion, secondary stresses stay low even on soft ground or in large temperature swings, and the design stays reliable. There is a construction advantage too: the two half-arches can be erected separately and finally connected at the crown hinge, which suits long-span erection.
Large-span roof structures: Gymnasiums, exhibition halls, hangars and station buildings often use three-hinged arches (as lattice members or tubular-steel trussed arches) for their wide roofs. They create a large column-free space with a low structural depth, carrying the roof self-weight and snow as a distributed load efficiently in compression. The hinged supports at both ends are robust against differential settlement of the foundations and reassuring for maintenance.
Historic stone and masonry arches: Roman aqueducts, stone bridges and church vaults have no explicit hinges, yet the locations where cracks form act as effective hinges, giving behaviour close to a three-hinged arch. Stability assessment of masonry uses "thrust-line analysis", checking whether the thrust line stays within the middle third of the arch section — and the parabolic thrust line in this tool is the basis of that idea.
Learning structural design and quick estimates: Because a three-hinged arch is statically determinate, the reactions can be worked out by hand from equilibrium alone, making it an ideal first topic for structural mechanics. In practice, before moving to a detailed indeterminate arch or a finite-element analysis, a determinate model like this tool gives the order of magnitude of the horizontal thrust and a first read on the size of the abutments and foundations. If an FEM result differs from this estimate by an order of magnitude, it is a sanity check that points to a boundary-condition mistake.
Common Misconceptions and Pitfalls
The most common misconception is designing the foundations from the vertical reaction alone and forgetting the horizontal thrust. Thinking of the supports with a beam mindset draws all the attention to the vertical reaction, but what really matters in an arch is the sideways horizontal thrust. Even under the default conditions, the thrust is 375 kN against a vertical reaction of 300 kN — it is not unusual for the horizontal component to be the larger one. Without an abutment, buttress or tie that resists this thrust, the supports slide outward and the arch collapses suddenly. Most arch failures originate not in the arch itself but in an inadequacy around the supports.
Next, the assumption that a parabolic arch has zero bending moment whatever the load. The bending moment is zero only when the arch axis coincides with the funicular (thrust line) of the load. A parabola is the thrust line specifically for a uniformly distributed load over the full span. Real bridges and roofs always carry asymmetric loads — snow on one side only, an offset live load, wind. Then the arch axis and the thrust line diverge and a bending moment appears. In three-hinged arch design these asymmetric load cases often govern the section, so do not feel safe with the uniform case alone.
Finally, the belief that the higher the rise, the better. It is true that raising the rise reduces the horizontal thrust inversely and lets the abutments be smaller. But a higher rise makes the arch axis itself longer, increasing the buckling risk of the compression member and the cost of the structural depth and the construction scaffolding. Conversely, making the arch too flat with a low rise generates a huge thrust from even a modest load and makes the support conditions severe. In practice the rise-to-span ratio is usually kept around 1/8 to 1/3 (h/L of about 0.12 to 0.33), balancing thrust, buckling, constructability and appearance.
How to Use
Enter span length (L) in meters and rise height (h) in meters to define parabolic arch geometry.
Apply uniformly distributed load (UDL) in kN/m across the arch span and optional point load in kN at crown.
Click Analyze to compute vertical reactions at supports, horizontal thrust H, resultant magnitude, and reaction angle; verify h/L ratio stability.
Worked Example
Concrete arch with span L=12 m, rise h=3 m, UDL=8 kN/m, crown load=20 kN. Rise-to-span ratio h/L=0.25. Vertical reaction at each support: 68 kN. Horizontal thrust H≈54.2 kN. Resultant reaction magnitude: 87.4 kN at reaction angle 51.3°. For comparison, a simply supported beam (no arch action) would require larger vertical support forces and zero horizontal restraint, demonstrating compression benefit of arch form.
Practical Notes
Three-hinge design (two supports + crown pin) eliminates indeterminacy; useful for masonry and concrete arches where thrust line control is critical.
Increase rise h to reduce horizontal thrust H significantly; h/L>0.3 favors arch action and reduces bending moments near crown.
Crown point loads create asymmetric thrust distribution; monitor resultant angle to ensure support bearings tolerate oblique reaction vectors.
Parabolic geometry assumes uniform UDL; for concentrated loads away from crown, use multiple crown load increments to approximate distributed effects.