A timber cord and arch is a roof structure where a curved laminated timber arch carries compression loads to the supports while a horizontal tension cord — usually a steel rod or a second timber member — ties the arch feet together to absorb the outward thrust. Otto Hetzer patented the glued laminated timber arch in Germany in 1906, which made long clear spans practical without buttresses. The arch handles vertical loads through axial compression along its centreline, and the cord prevents the supports from spreading. Modern glulam tied-arch roofs span 30 to 100 m over churches, gymnasiums, and aircraft hangars.
Timber Cord and Arch Interactive Calculator
Vary span, rise, roof load, and tie stress to see arch thrust, reactions, compression, and required tension-cord area.
Equation Used
For a parabolic tied arch carrying a uniform vertical line load, the horizontal thrust is H = wL^2/(8r). The tension cord must resist this thrust, while each support carries a vertical reaction V = wL/2. The springing compression is the vector combination of H and V, and the tie area shown is based on the entered allowable tie stress.
- Parabolic three-hinged tied arch under uniform vertical load.
- Horizontal cord carries the full outward thrust as axial tension.
- Load is entered as total line load per metre along the span.
- Bending effects, connection eccentricity, buckling, and load factors are not included.
Operating Principle of the Timber Cords and Arches
A timber arch works because a curved member loaded from above wants to flatten — and flattening pushes the feet outward. If you let the feet move, the whole arch collapses. Tie the feet together with a cord and that outward thrust becomes pure tension in the cord, leaving the arch in nearly pure compression. That is the entire trick. Timber happens to be excellent in compression along the grain (around 30-45 MPa for spruce glulam) and decent in tension when you reinforce it with steel, so the geometry plays to the material's strengths.
The shape matters more than most builders realise. For a uniformly distributed load like snow on a flat roof, the ideal arch shape is a parabola — the thrust line stays inside the centreline of the arch and bending moments stay near zero. Deviate from that shape and the arch picks up bending stress, which timber resists far less efficiently than compression. A typical glulam arch holds the centroid within ±L/200 of the theoretical thrust line, where L is the span. Push that tolerance wider and you get visible sag, splitting along glue lines, or worse — sudden buckling between purlins.
Rise-to-span ratio is the next big lever. A shallow arch (rise/span ≈ 1/8) generates massive horizontal thrust at the cord — sometimes 3 to 5 times the vertical reaction — which means a fat tension chord and serious anchorage. A steep arch (rise/span ≈ 1/3) cuts thrust dramatically but eats headroom and adds material. The classic three-hinged timber arch lands around 1/5 to 1/6 because that hits the sweet spot between cord force and clear-height. Get the rise wrong and you'll either oversize the tie rod by a factor of two or lose useful floor volume to wasted curvature.
Key Components
- Glulam Arch Rib: The primary curved compression member, built from 33-45 mm thick laminations of spruce, Douglas fir, or southern yellow pine bonded with melamine-urea-formaldehyde or resorcinol adhesive. Cross-sections typically run 140-240 mm wide and 400-1200 mm deep depending on span. The lamination thickness must stay under 45 mm so the steam-bending or cold-bending radius does not crack the outer fibres.
- Tension Cord (Tie Rod): A horizontal member connecting the two arch feet that absorbs outward thrust. For spans under 25 m this is usually a 25-50 mm diameter Grade 8.8 steel rod with turnbuckles for tensioning. For longer spans or exposed installations it becomes a glulam tension chord with steel reinforcement. The cord must be tensioned to roughly 5-10% above the calculated dead-load thrust to prevent sag and rattle.
- Foot Connection (Shoe): A welded or cast steel shoe that transfers compression from the arch into the support and tension into the cord. Pinned shoes allow rotation, which makes the structure a three-hinged arch — statically determinate and far more tolerant of foundation settlement. Bolt slop above 0.5 mm at the pin causes audible creak under wind load and accelerates wear at the bearing surfaces.
- Crown Hinge or Splice: A pinned joint at the top centre of the arch where the two halves meet. This third hinge makes the arch insensitive to thermal expansion and minor differential settlement. A typical crown hinge uses a 30-60 mm diameter pin in a steel knife-plate connector, machined to H7/h6 fit so the rotation stays smooth over the 50+ year design life.
- Purlins and Decking: Secondary members spanning between arches at 1.2-2.4 m centres, carrying roof deck, insulation, and snow load into the arch ribs. Purlins also brace the arch laterally — without them, a deep narrow arch will buckle out of plane long before it fails in compression.
Where the Timber Cords and Arches Is Used
Timber cord and arch structures show up wherever a builder needs a long clear span with no internal columns, a warm exposed ceiling finish, and reasonable cost. The combination of high strength-to-weight ratio, fire resistance through charring, and the visual warmth of exposed glulam makes the system dominant in public assembly buildings. Cord-and-arch roofs handle 30-100 m spans routinely, and the longest documented glulam tied-arch — the Richmond Olympic Oval in British Columbia — clears 100 m using a hybrid timber-steel system. The mechanism fails most often not from material overload but from cord anchorage corrosion or lateral buckling between inadequate purlin ties.
- Sports Facilities: The Richmond Olympic Oval in British Columbia uses a glulam tied-arch roof spanning 100 m over the speed-skating track, built with Douglas fir glulam ribs and a steel tension network.
- Religious Buildings: St. Mary's Cathedral in Tokyo by Kenzo Tange uses laminated timber arches with embedded tension cords to clear the central nave without columns.
- Aircraft Hangars: WWII-era American hangars at Naval Air Station Tillamook in Oregon used 90 m clear-span timber arches with steel tension cords — Hangar B is now a museum and the structure still stands.
- Equestrian and Agricultural: Riding arenas built by manufacturers like Powerlam and Cover-All use 25-40 m glulam arches with internal steel tie rods to clear the riding surface without posts.
- Public Markets and Stations: The Centre Pompidou-Metz in France uses a curved laminated timber lattice acting as a continuous arch-cord system, woven from over 1,800 glulam beams.
- Schools and Gymnasiums: Mid-20th-century school gymnasiums across the American Midwest and Canada commonly used Timber Structures Inc. or Unit Structures bowstring trusses — a tied-arch variant — clearing 18-30 m over basketball courts.
The Formula Behind the Timber Cords and Arches
The horizontal thrust at the arch foot is what sizes the tension cord, the shoe connection, and the foundation anchorage. Get this number wrong and either the cord stretches and the arch sags, or you over-spec the tie rod by a factor of two and waste steel and money. The formula relates uniformly distributed load, span, and rise — and the rise is the variable you actually control during design. At a shallow rise of L/10 the thrust runs near 1.25× the total vertical load, which is brutal on the cord. At a typical rise of L/5 the thrust drops to about 0.625× the vertical load — the sweet spot for most glulam arch builds. Push to a steep L/3 rise and thrust falls to roughly 0.375× the vertical, but the arch length and material cost climb sharply.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| H | Horizontal thrust at the arch foot (cord tension) | kN | lbf |
| w | Uniformly distributed load along the span | kN/m | lbf/ft |
| L | Span between arch supports | m | ft |
| r | Rise of the arch from spring line to crown | m | ft |
Worked Example: Timber Cords and Arches in a curling rink glulam tied-arch roof
A curling club in thunder bay ontario is sizing the tension cord on a new 36 m clear-span glulam tied-arch roof over an 8-sheet curling rink. The roof carries a combined dead and snow load of 3.2 kN/m² and the arches sit at 6 m centres. The architect wants to evaluate three rise options — 3.6 m (L/10), 7.2 m (L/5), and 12 m (L/3) — to pick the cord size and ceiling height combination.
Given
- L = 36 m
- Tributary width per arch = 6 m
- Total surface load = 3.2 kN/m²
- w = 19.2 kN/m
Solution
Step 1 — compute the line load on a single arch by multiplying the surface load by the 6 m tributary width:
Step 2 — at the nominal L/5 rise (r = 7.2 m), compute the horizontal thrust at the cord:
Step 3 — at the shallow L/10 rise (r = 3.6 m), the cord force doubles:
That is enormous — you would need a 60 mm Grade 8.8 tie rod or a fully reinforced glulam tension chord, and the foundation anchorage becomes a serious problem. The shallow profile also makes the arch sensitive to non-uniform snow drift, which pushes the thrust line off the centreline and induces bending the timber does not handle well.
Step 4 — at the steep L/3 rise (r = 12 m), the cord force drops sharply:
Now a 40 mm tie rod handles it comfortably, but the arch crown sits 12 m above the spring line — that is a 4-storey-tall ceiling void above a curling rink, which adds heating cost and roof material. The L/5 nominal at 432 kN sits in the sweet spot: a 50 mm Grade 8.8 rod with turnbuckles handles the load with margin, and the 7.2 m crown height is reasonable for the building.
Result
Nominal cord tension at the L/5 rise is 432 kN — well within the capacity of a single 50 mm Grade 8. 8 tie rod tensioned to roughly 460 kN service load. The shallow L/10 option doubles the cord force to 864 kN and forces a much heavier rod plus reinforced anchorage, while the steep L/3 option cuts cord force to 259 kN but burns 12 m of vertical headroom. If your installed cord measures higher tension than the 432 kN prediction during dead-load lift-off, suspect three causes in order: turnbuckle over-tightening past the 5-10% pre-stress allowance, foundation spread of more than 5 mm at the shoes letting the geometry flatten, or asymmetric snow drift loading the arch off-axis and shifting the thrust line. Any of these will show up as visible cord sag, audible creaking at the foot shoes, or hairline checks along the inner glulam laminations near the haunch.
Choosing the Timber Cords and Arches: Pros and Cons
A timber tied-arch is one of several ways to clear a long span without internal columns. The right choice depends on span, ceiling aesthetic, fire performance, and budget. Here is how the timber cord-and-arch stacks up against the two most common alternatives — steel portal frames and reinforced concrete arches.
| Property | Timber Cord and Arch (glulam) | Steel Portal Frame | Reinforced Concrete Arch |
|---|---|---|---|
| Typical clear span | 20-100 m | 15-60 m | 30-200 m |
| Cost per m² of roof (relative) | 1.0× | 0.7-0.9× | 1.4-2.0× |
| Self-weight per m² of structure | 35-60 kg/m² | 25-45 kg/m² | 180-300 kg/m² |
| Fire resistance (untreated) | 60-90 min via charring | 10-20 min before yield | 120+ min |
| Erection time on site | Fast — pre-fab arches lift in days | Fastest — bolted in hours | Slowest — formwork and cure weeks |
| Lifespan in dry indoor service | 80-100+ years | 60-80 years | 100+ years |
| Sensitivity to moisture | High — needs envelope protection | Medium — corrosion management | Low |
| Ceiling aesthetic | Warm exposed timber finish | Industrial — usually clad | Heavy — usually clad |
Frequently Asked Questions About Timber Cords and Arches
The cord tension being correct only proves the global thrust is right — it does not prove the thrust line is following the arch centreline. Hairline checks at the haunch (the curved transition between the leg and the rib) almost always mean the local thrust line has migrated inside the centroid, putting the inner fibres in tension. Timber pulls apart on tension-perpendicular-to-grain at stresses as low as 0.5-1.5 MPa, far below its compression strength.
Three usual culprits: non-uniform load (heavy snow drift on one side), foundation settlement of 10+ mm at one shoe shifting the geometry, or — most commonly — the original arch shape was specified as a circular arc instead of a parabola. A circular arc only matches the thrust line at one specific load case; everything else induces bending. Fix is usually to add a thrust-line-correcting steel reinforcement strap at the haunch, not to retension the cord.
Use three-hinged. The crown hinge makes the structure statically determinate, which means it tolerates differential foundation settlement, thermal expansion, and timber shrinkage without inducing secondary stresses. For a 40 m span, summer-to-winter thermal cycling alone can produce 8-12 mm of length change in the cord, and a two-hinged arch will lock that movement into bending stress at the crown.
Two-hinged arches make sense only for spans under about 25 m where thermal effects are small, or where the architectural profile cannot accept a visible crown joint. The crown hinge on a three-hinged arch is also where you splice the two halves for transport — glulam arches longer than about 25 m are nearly impossible to truck in one piece, so the hinge solves a logistics problem at the same time.
That is a 3× deviation, which almost never comes from material modulus error. The two likely causes: cord anchorage slip and foot-shoe bolt slop. A 25 mm clevis pin in a 25.5 mm hole gives 0.5 mm of slop per connection — multiplied across two cord ends and two arch feet that is 2 mm of geometric slack before the cord even picks up load. Add bolt-bearing crush in the timber at the shoe (timber yields locally at around 7-10 MPa bearing perpendicular to grain) and you can easily lose another 10-15 mm.
Diagnostic check: re-tension the cord with the building unloaded, mark the turnbuckle position, then load the roof and re-measure. If the turnbuckle has rotated, the slip is in the cord ends. If it has not, the slip is in the foot shoes — check for ovalised bolt holes or crush at the bearing plates.
Yes, but the cross-section gets surprising fast. Glulam pulls apart at about 16-24 MPa parallel to grain (much less than steel's 400-800 MPa). For the 432 kN cord force in the worked example, you would need a glulam tension chord around 200 × 400 mm — versus a 50 mm steel rod doing the same job. That is a visually heavy member running across your ceiling.
The other catch is connection. End-grain timber tension splices are weak, so the chord typically has steel side-plates with through-bolts or glued-in steel rods — and that connection often becomes the weakest link, capping the chord at maybe 60-70% of the gross-section capacity. Most architects who want the timber-only look end up with a hybrid: glulam chord with a concealed steel rod down its centreline taking the actual tension.
The cord is acting as a long taut string and the wind is exciting its first vibration mode — same physics as a guitar string. For a 36 m span at typical pre-tension, the natural frequency lands around 2-4 Hz, which overlaps with gust frequencies in moderate wind. The rattle you hear is the cord slapping against intermediate hangers or the turnbuckle housing on each cycle.
Fix is one of three things: increase pre-tension by 20-30% to push the natural frequency above the gust band, add a damper or neoprene isolator at the intermediate cord hangers, or fit a lightweight stay cable mid-span to break the vibration mode into two shorter half-spans. Increasing pre-tension is cheapest but watch the foundation anchorage capacity — you cannot exceed it.
Bowstring truss wins on material cost — the lattice web of small members between the curved top chord and the straight bottom chord uses 30-40% less timber than a solid glulam rib doing the same job. It is the historical default for spans 18-30 m, which is why every mid-century farm building and small gymnasium in North America has one.
Tied-arch (solid glulam rib) wins on aesthetics, fire resistance, and load path simplicity. The clean curve reads as architecture; the lattice reads as utility. For fire performance, a solid glulam rib chars predictably at 0.7 mm/min and retains its load capacity for 60-90 minutes, while a lattice with smaller members loses capacity faster because each lattice member has higher surface-area-to-volume ratio. For a 25 m public-assembly building with exposed structure, pick the tied-arch. For a 25 m agricultural or industrial building where structure is hidden above an insulated ceiling, the bowstring is more economical.
References & Further Reading
- Wikipedia contributors. Glued laminated timber. Wikipedia
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