Stone arch bridge: load it and watch how it carries weight

What the model shows

Walk under an old stone bridge and look up. No mortar is strong enough to glue the stones together against gravity, and yet the arch has stood for centuries. A steel railway bridge nearby looks completely different, a lattice of thin bars, and a suspension bridge further along the coast hangs its road from two cables. All three solve the same problem: carrying a load across a gap and down into the ground on either side.

The model puts the three side by side. On the left is a semicircular stone arch, in the middle a Pratt truss, and on the right a suspension bridge with two towers and two anchorages. Each is drawn to its own scale, so the arch is about 8 m across, the truss 30 m and the suspension bridge 160 m between its towers. The "Load position" and "Load" sliders move a weight across all three at once, and the panel in the corner reports the forces worked out for that position. The "Stones laid" slider and the "Centring" switch let you build the arch and take away its timber support.

Why stone can only push

Stone is very strong when it is squeezed and very weak when it is pulled. A stack of blocks with no mortar cannot carry any tension at all: the joints simply open. So a stone bridge has to be shaped so that every joint is pressed together and none is pulled apart.

An arch does that by turning the downward weight of the stones and the traffic into a sloping push. Each wedge-shaped stone, called a voussoir, leans on its neighbours, and the push travels round the curve and down into the abutments. Because the push slopes, the abutments have to resist two things: the weight coming down and a sideways shove called the thrust. Take away the sideways support and the feet of the arch spread while the crown drops.

Robert Hooke described the right shape in 1675 with a Latin anagram. Decoded, it says that a hanging chain, turned upside down, gives the shape of a rigid arch. A chain can only be in tension, so it hangs in whatever shape puts every link in pure tension. Flip that shape over and every link would be in pure compression instead, which is exactly what stone wants.

Building on centring, and why the keystone matters

Half an arch cannot stand. Each stone leans inward, and with nothing on the other side to lean against, it would rotate and fall. So builders first put up a timber frame called centring, with its top shaped like the underside of the arch, and lay the stones on it from both springings toward the middle.

The last stone at the crown is the keystone. Once it is pushed in, the two halves lean on each other and the ring is closed. The builders then knock out wedges at the tops of the centring posts, the frame drops a few centimetres, and the arch takes its own weight for the first time. This is called striking the centring. In the model, set "Stones laid" below 15 and turn the centring off: the stones fall. Lay all 15 first and the same switch leaves the arch standing.

The keystone's shape is nothing special here. In the model it is the same size and weight as the other fourteen stones. What makes it the keystone is that it goes in last.

The thrust line, in numbers

The model ring has a centreline radius of 4 m, a thickness of 0.6 m and a width of 3 m. With granite at 23.5 kN per cubic metre, the whole ring weighs π × 4 × 0.6 × 3 × 23.5 = 532 kN, about 54 tonnes, split into 15 stones of 35 kN each.

Treat each stone as a single weight acting at its centre of gravity and pick a horizontal thrust H. Starting at one springing, the push runs in a straight line until it meets the first weight, then bends downward by an amount set by that weight, then runs straight to the next one, and so on. The bent line that results is the thrust line. Its slope on each stretch is the vertical force still to be carried divided by H, which is the same rule that sets the shape of a hanging chain.

The arch can settle on its own H within limits. A small thrust gives a deep, pointed thrust line, and a large thrust gives a flatter one. Jacques Heyman's safe theorem says that if any one of these lines can be drawn inside the stonework at every joint, the arch will not collapse under those loads, provided the stones neither crush nor slide. The model searches all possible lines and draws the one that keeps furthest from the edges. The percentage in the panel is how far that line strays from the middle of the ring at its worst joint, as a fraction of half the thickness. Under its own weight the model ring reaches 71%, so there is room to spare. Each abutment then pushes back with H = 104 kN sideways and 266 kN upward, half the ring's weight.

Put a load on the ring and the line has to bend sharply under it. A load of about 100 kN at the crown uses up all the room: no line fits any more. The worst place in this model is just beside the crown, at 45% of the span, where 72 kN is enough. At a quarter of the span the limit is 129 kN, and close to the springings it is more than 500 kN, because the load there goes almost straight down into the abutment. The joints where the line touches the edge are where the arch would crack open and turn like a hinge; four such hinges turn the ring into a mechanism and it falls. In the model those joints turn red.

A bare ring is the worst case. Real stone bridges carry earth fill and a road above the ring. That fill adds weight, which helps hold the thrust line in place, and it spreads a wheel load over a wide part of the ring.

How a truss turns bending into push and pull

A truss replaces a solid beam with triangles of straight bars joined at their ends. If the joints are treated as pins and the loads arrive only at the joints, each bar carries a single force along its length: compression, shown in blue, or tension, shown in red.

The forces come from the method of joints. At each pin, the forces from the bars and any load or support reaction must add up to zero, both horizontally and vertically. The model truss has 12 joints, so it gives 24 equations, and it has 21 bars plus three support reactions, so there are exactly 24 unknowns. Solving them together gives all the forces at once.

Put a 200 kN truck in the middle. Each of the two trusses takes half, 100 kN, and each support pushes up with 50 kN. Cut the truss through the middle panel and take moments: the bending moment at midspan is 50 kN × 15 m = 750 kN·m, and with the chords 5 m apart, the top chord carries 750 ÷ 5 = 150 kN of compression. The bottom chord in the same panel carries 100 kN of tension, and each diagonal carries 50 ÷ sin 45° = 71 kN of tension. The panel shows these same values.

Why diagonals change sign under a moving truck

In a Pratt truss the diagonals slope down toward the middle. That arrangement keeps them in tension under a symmetric load, which suited 19th-century builders who could make long iron bars cheaply but feared that slender bars in compression would buckle. Thomas and Caleb Pratt designed it in 1844.

A truck does not stay in the middle, though. Drive it to one third of the span and the shear in the middle panel reverses direction. Diagonal U2L3, which carried 71 kN of tension, now carries 47 kN of compression. Engineers handle this with an influence line, a plot of one bar's force as the load moves across, and check each bar for the worst value in both directions. Older Pratt trusses with tension-only diagonals added crossing counters in the middle panels for this reason.

Why a hanging cable makes a parabola

A suspension bridge deck hangs from vertical hangers, and in the model there is one every 8 m. The deck's weight, 40 kN per metre for each cable, arrives at the cable as equal loads at equal spacing, 320 kN at each hanger.

A cable cannot resist bending, so its sag at any point must equal the bending moment that a simple beam of the same span would have under the same loads, divided by the horizontal pull H. For a uniform load the moment diagram is a parabola, so the cable is a parabola too. At midspan the moment is wL² ÷ 8, and since the sag there is f, H = wL² ÷ (8f) = 40 × 160² ÷ (8 × 16) = 8,000 kN. Near the towers the cable also carries the vertical share of the deck weight, so the tension there rises to 8,558 kN.

A chain hanging under its own weight forms a catenary instead, because its weight is spread along the curved length rather than evenly along the span. For a 160 m span with 16 m of sag, the two curves differ by 52 mm at most. A real cable carries both its own weight and the deck, so its shape lies between the two.

A truck breaks the pattern. With no stiffening girder in the model, a 300 kN truck (150 kN per cable) at a quarter of the span pulls a kink into the cable. Because the cable does not stretch, the rest of it rises a little, H climbs to 8,213 kN, and the cable under the truck drops by 0.24 m. The model draws that movement ten times larger so you can see it. Real suspension bridges use a stiff deck girder to spread such loads over many hangers.

Towers and anchorages

The cable runs over a saddle on each tower and down the back span to an anchorage. If the saddle can roll, the horizontal pull is the same on both sides, so the towers mostly carry a vertical load. In the model each backstay drops 16 m over a 50 m back span. Its tension is H divided by the cosine of that slope, 8,400 kN under the deck alone, and it pulls upward on the anchorage with 2,560 kN. That is why anchorages are such large blocks: their weight holds the cable end down.

Korean bridges built on these ideas

Korea's surviving stone arch bridges are called hongyegyo or hongyo, which means rainbow bridge. Seungseongyo at Seonamsa temple in Suncheon is a single semicircular arch of about 140 stones. According to the National Heritage Service it was first built in 1707 and repaired after a flood in 1713, and it was designated a Treasure in 1963. Beolgyo Hongyo in Boseong was built in 1729 and rebuilt with three arches in 1737; it is the largest of the stone rainbow bridges still standing. Both have a carved dragon head projecting under the crown.

Among suspension bridges, Namhae Bridge, opened on 22 June 1973 with a total length of 660 m, was Korea's first suspension bridge. Yi Sun-sin Bridge between Yeosu and Gwangyang has a main span of 1,545 m between its towers; it opened temporarily in May 2012 and fully on 8 February 2013.

Worked example: how a flatter arch pushes harder

Take the model arch between its springings: a span of L = 8 m and a rise of f = 4 m, which is the 4 m radius. Put 100 kN on the crown and treat the arch as three-hinged, with a hinge at each springing and one at the crown. By symmetry each abutment carries 50 kN upward. Now cut the arch at the crown hinge and take moments about it for the left half: the 50 kN reaction acts 4 m from the crown, and the horizontal thrust H acts 4 m below it. So 50 × 4 = H × 4, and H = 50 kN. In general H = PL ÷ (4f) = 100 × 8 ÷ 16 = 50 kN.

Keep the span and the load but flatten the arch. At f = 2 m the same formula gives H = 100 kN, and at f = 1 m it gives 200 kN. The vertical reaction stays at 50 kN every time. Only the sideways push changes, and it is inversely proportional to the rise. A flat arch does not need stronger stone so much as stronger abutments.

The same rule holds for a uniform load w. A parabolic arch carrying w per metre has H = wL² ÷ (8f), which is exactly the cable formula turned upside down. The Zhaozhou Bridge in Hebei, completed in 605, makes the contrast concrete. The American Society of Civil Engineers gives it a span of 37 m and a rise of only 23.9 ft, about 7.3 m. Treating both it and a semicircular arch of the same span as parabolas under a uniform load (an idealisation), the thrust per unit of total load is L ÷ (8f). For Zhaozhou that is 37 ÷ (8 × 7.3) = 0.63. For a semicircle, whose rise is half the span, it is 0.25. The low arch pushes sideways about 2.5 times harder for each kilonewton it carries, which is the price of a gentle road profile.

Centring, ring by ring

The "Stones laid" slider counts from 0 to 15, and each stone in the model weighs about 35 kN. At 10 stones the timber centring is carrying roughly 350 kN of masonry that has nothing else to lean on. Turn "Centring" off at that point and the stones fall, because an incomplete ring is not a closed structure and no thrust line can be drawn from one springing to the other.

At 15 stones the situation changes completely. The ring is closed, the model finds a thrust line inside it, and the centring carries nothing useful. Builders lower it in stages rather than knocking it away. The Brick Industry Association's technical note on masonry arches treats easing the centering as a step that comes only after the keystone is placed.

When that happens the arch does not jump to one fixed state. The horizontal thrust can settle anywhere in a range. Block, DeJong and Ochsendorf report, for a semicircular arch with a ring thickness of 18% of its radius, a minimum thrust of about 16% of the arch's weight and a maximum of about 24%. The model ring is slightly thinner, 0.6 m on a 4 m radius or 15%, so its range is somewhat narrower. The thrust the model draws, 104 kN against a ring weight of 532 kN, is 19.5% of the weight and sits comfortably in the middle of that kind of range. If an abutment spreads a little, the thrust moves toward the lower end of its range, and the arch keeps standing as long as some line still fits inside the ring.

Arch, truss and cable compared

The three structures answer the same question in three different ways, and the clearest difference is what each one does to its supports.

• Stone arch. The ring is always in compression. Each abutment is pushed down and outward. In the model: 266 kN down and 104 kN out under the ring's own weight.
• Pratt truss. The truss only presses straight down on its supports. Its bars take compression or tension depending on where the load stands, and the chords supply the bending resistance. In the model the chord force for a 100 kN load is the midspan moment divided by the depth: 750 ÷ 5 = 150 kN. A truss of half the depth, 2.5 m, would need chords carrying 300 kN for the same load.
• Suspension bridge. The cable is always in tension and pulls the supports inward and upward. Its horizontal pull is H = wL² ÷ (8f), so halving the sag doubles it: the model's 8,000 kN would become 16,000 kN at 8 m of sag.

For a uniform load, compare the horizontal force with the load carried. The model cable has H ÷ (wL) = 160 ÷ (8 × 16) = 1.25, because its sag is a tenth (1/10) of its span. A semicircular arch has a ratio of 0.25 in the same idealisation, because its rise is half the span. A cable that sags a tenth (1/10) of its span pulls five times as hard sideways as the arch pushes, which is why anchorages and backstays are so massive.

The materials follow the same logic. Stone and cast iron are strong when squeezed and brittle when pulled, so they suit arches. Steel can take both, so it suits trusses. A cable of steel wire is useless in compression but excellent in tension, so it suits a suspension bridge. The Iron Bridge at Coalbrookdale in Shropshire shows how habits carry over: English Heritage describes a single cast-iron span of 30 m made of five semicircular ribs, joined with dovetail and shouldered joints borrowed from carpentry.

Real bridges and what they test

Pont du Gard, France. UNESCO describes a Roman aqueduct bridge of three tiers of arches, rising to nearly 48.77 m, with the lower two levels built of large stone blocks. In a row of equal arches each one pushes sideways on its neighbours, and the pushes cancel at every pier except the last. Only the two end abutments have to resist a net thrust.

Zhaozhou Bridge, China, 605. The ASCE landmark record calls it the world's oldest open-spandrel arch bridge. It was built by Li Chun from 28 thin curved limestone slabs joined with iron dovetails, and small arches at the ends of the deck pass part of the load to the main arch and let flood water through. Its low rise follows from the thrust calculation above.

Iron Bridge, England, 1779. English Heritage records 378 tons of iron and a cost of about £6,000 against a first estimate of £3,200. The BBC's history page counts 482 main castings and says the first arch spanned the river on 2 July 1779. English Heritage adds that 70% of the components were cast individually to fit, so no two are quite the same. The BBC page adds that the south abutment moved severely and was demolished in 1802, a reminder that an arch is only as reliable as the ground that takes its thrust.

Golden Gate Bridge, 1937. The ASCE landmark page gives a main span of 4,200 ft (about 1,280 m), the longest single span in the world when it was built, with construction starting in November 1932. That is eight times the model's 160 m. As an illustration of scaling only, keep the model's sag ratio of 1/10 and its load of 40 kN per metre of cable. The horizontal pull would then grow in proportion to the span, from 8,000 kN to 64,000 kN. The real bridge has different loads and a different sag, so this is a scaling check, not a figure for the bridge.

Tacoma Narrows Bridge, 1940. The original design had 25 ft stiffening trusses. A cheaper design with 8 ft solid plate girders replaced it, and the bridge opened on 1 July 1940. ASCE records that it failed on 7 November 1940 in a wind of only 42 mph, four months after opening. The cause was aerodynamic, which the model does not include at all. The deck was extremely shallow for its width, a depth-to-width ratio of 1 to 72, so it twisted easily. The 1950 replacement used a 33 ft stiffening truss and openings for wind to pass through the deck.

Common misconceptions

• "The keystone holds the whole arch up." Every stone is squeezed by its neighbours, and the keystone is one of them. In the model it has the same size and weight as the other 14. What matters is that it closes the ring.
• "A load at the crown is the worst case." In the model it is not. The ring tolerates about 100 kN at the crown but only 72 kN at 45% of the span, and 129 kN at a quarter. The worst position is where the load forces the thrust line furthest from the middle of the ring.
• "More bars make a truss stronger." Bars only help if they carry force. In the model the verticals nearest the ends carry zero force unless the load stands on their joint, and a diagonal can swap from tension to compression as the truck moves.
• "A hanging cable is a catenary." Only when the cable's own weight dominates. With a deck much heavier than the cable, the shape is a parabola, and in the model the two curves differ by at most 52 mm over 160 m.
• "Tacoma Narrows was resonance." The American Physical Society points out that this textbook explanation is incorrect. The Federal Works Administration found resonance an improbable cause, and the accepted explanation is torsional flutter, a self-excited twisting in which the deck's own motion drew more energy from the wind.

Things to try on the screen

1. Use the scene buttons at the top of the stage: stone arch, truss, suspension bridge, real bridges, or all three side by side. Each bridge fills the stage on its own.
2. On the stone arch, drag the cast-iron weight along the ring. The green line follows it, and the dots running down the line show the push travelling to the abutments. Around 45% of the span, 75 kN is already enough to push the line out of the ring.
3. Tap any stone in the ring. It slides out and the whole ring falls, the keystone included. Press "Put the stones back" to rebuild it.
4. Set "Stones laid" to 10 with the centring on, then switch "Centring" off. The stones fall. Lay all 15 and switch it off again: the arch stands.
5. With no load, lower "Ring thickness" from 0.60 m. Below about 0.43 m no thrust line fits inside the ring any more.
6. On the truss, drag the truck. With 200 kN in the middle the top chord carries 150 kN of compression. Move it to one third of the span and diagonal U2L3 turns from red to blue.
7. On the suspension bridge, set "Load" to 300 and drag the truck to a quarter of the span. The cable kinks and drops 0.24 m, drawn ten times larger.
8. Open the real bridges: Zhaozhou, Brooklyn, Golden Gate and Akashi Kaikyō, each drawn to its measured span and height.
9. The Controls tab lists 11 challenges, from breaking the arch with 100 kN or less to finding the 7 hidden things in the scenes.

Where the model is simpler than a real bridge

• The arch is a bare ring with no fill, no spandrel walls and no road. Joints are radial, the stones are rigid, and crushing and sliding are ignored, as in Heyman's assumptions.
• The arch load is a single vertical point load resting on one stone.
• The truss has pin joints, ignores its own weight and the deck's, and treats the truck as one point load shared between the two nearest floor beams.
• The suspension deck has no bending stiffness, the cable does not stretch, the backstays are straight and the tower saddles roll freely. Cable and deck movements are drawn ten times larger.

How the numbers were checked

The statics were also worked through by hand. For the truss with 100 kN at midspan the hand values match the reactions, the 150 kN top chord, the 100 kN bottom chord, the 70.7 kN end posts and diagonals, and the zero force in the verticals next to the ends, and it confirms that all 12 joints balance for 61 truck positions. For the arch they match H = PL ÷ (4f) for a three-hinged arch with a crown load, and for a semicircular arch with t/R = 18% the thrust range comes out of 16% to 24% of the arch's weight, against 16% and 25% reported by Block, DeJong and Ochsendorf. For the cable they confirm that the hanger points lie on y = 4fx(L − x) ÷ L² and that H = wL² ÷ (8f).