What a taper actually removes
Strength in a leg is governed by the section modulus, and the section modulus of a square section falls with the cube of the side. Take a leg that is 38 mm square under the seat and 22 mm square at the floor. The cross-sectional area falls from 1444 to 484 square millimetres, which is 66 percent of the material gone. The section modulus falls from 9153 to 1775 cubic millimetres. That is 19 percent of the original bending strength, at the point where the leg meets the floor and takes every kick, every chair drag and every uneven tile.
That number is not a defect, it is the design. A chair leg carries very little of the guest load in pure compression, because the load path is nearly vertical and the leg is short. What it carries is bending from side loads: a guest pushing back, a chair dragged across a floor, a table edge pressed against a knee. Bending demand at the floor is close to zero because there is no moment arm below the contact point, and it grows toward the seat. A taper that removes material at the bottom and keeps it at the top follows the bending moment diagram, which is why every mid-century chair in production uses one.
The mistake is tapering too far or starting too high. A 38 to 22 mm taper over the last 300 mm of a 450 mm leg is a 5.3 degree taper and keeps the section at the seat intact. The same 22 mm floor dimension reached over the full 450 mm, or a 20 mm floor dimension, removes material where the joint needs it and turns a designed taper into a fatigue failure. The ratio we hold on wood legs is a floor section no smaller than 0.58 of the seat section: 22 mm from 38 mm is 0.58, 20 mm from 38 mm is 0.53 and needs a stretcher to compensate. On the JP-C012 wood dining chair the front legs run from 38 mm under the seat to 23 mm at the floor over the lower 320 mm, which is a 0.61 ratio and stays just inside that line with 130 mm of full section left above the taper for the joint.
Splay: stability bought with joint moment
Splay is the angle between the leg and vertical, and it does two opposite things at once. It widens the base, which is the single most effective way to stop a chair tipping: a 450 mm leg splayed 6 degrees moves the foot 47 mm outward, which widens the support polygon by 94 mm front to back and cuts the tipping tendency under a backward lean by roughly a fifth. It also inclines the leg, so the vertical load no longer runs down the leg axis and the joint at the seat has to resist a bending moment it did not resist before.
The moment is small at low angles and grows faster than the angle suggests. At 3 degrees the eccentricity of a 150 kg load is negligible against the joint capacity. At 6 degrees it is around 5 percent of the leg length, and at 10 degrees it approaches 18 percent, at which point the joint is doing structural work rather than locating work. That is why splay above 8 degrees almost always comes with a stretcher between front and back legs, and why a chair with 10 degrees of splay and no stretcher will be the first one in the room to loosen.
There is a second effect that shows up in the room rather than in the calculation: splayed legs concentrate load on the outer edge of the foot, so the glide wears on one side and the leg tips slightly. Over 24 months that produces a rocking chair on a flat floor, and the rocking loads the joint in a direction the design never tested. A 3 mm radius on the foot and a glide with a 1 mm wall instead of 0.6 mm removes most of it for a few cents per chair.
Splay is also where the aesthetic and the engineering argue. The visual lightness that designers ask for comes from both taper and splay, and the cheapest way to deliver it is to increase both. The specification that survives service is usually 4 to 6 degrees of splay with a 0.6 taper ratio, not 9 degrees with 0.5.
The shoulder carries the load, not the tenon
Almost every tapered leg failure we see starts as a hairline gap at the shoulder, not as a broken tenon. The geometry explains why. A leg joint is a mortise in the leg, a tenon on the seat rail, and a shoulder on the rail that lands on the leg face. The tenon locates and the glue line around the tenon resists pull-out; the shoulder is what transfers compression and bending into the leg face. When the shoulder is 2 mm instead of 5 mm, or when the rail is machined 0.3 mm short, the joint closes on the tenon alone and the tenon becomes a lever working inside the mortise.
The numbers on a typical side rail are a 10 mm thick tenon, 35 to 40 mm deep, in a 22 to 25 mm rail, with a 5 mm shoulder all round. That gives 700 to 800 square millimetres of glue area on the tenon cheeks and a shoulder that seats across the full rail width. Drop the tenon to 8 mm and the cheeks lose 20 percent of their area; drop the shoulder to 2 mm and the joint loses the bearing surface that stops rotation under a side load of 300 to 400 newtons, which is an ordinary push on the seat edge.
Glue is not a substitute for fit. A polyvinyl acetate glue line on a hard hardwood reaches 8 to 12 MPa in shear on a good joint and much less on a starved one, and a 0.2 mm gap is enough to starve it. The reason factory joints outperform site repairs is not the glue, it is that the mortise and tenon are machined to 0.1 mm and the shoulder is cut in the same setup, so the joint closes metal to metal before the glue cures.
This is the detail that separates the curved back dining chairs that last from the ones that do not: the curve in the back changes the rail angle, and if the shoulder is not recut to that angle the joint seats on one edge only. A curved back is not harder to build, it is harder to build without measuring.
The 18 month wobble: moisture, glue creep and load cycles
Timber moves with humidity, and a chair leg moves in two directions at once. A 38 mm leg at 9 percent moisture content will sit at 12 to 13 percent after two humid months in a coastal city, and the tangential movement over that swing is around 0.3 percent per point of moisture, which is roughly 0.3 mm across the leg. That sounds trivial until you consider that the joint was machined to a 0.1 mm fit. The leg swells, the mortise closes on the tenon, the shoulder lifts, and the glue line that was carrying shear starts carrying peel.
The second mechanism is glue creep under sustained load. A chair in a dining room is loaded 200 times a day for 12 hours and carries a static load of 60 to 90 kg when occupied. Over 18 months that is well past a million load applications on the front legs of the busiest four chairs in the room, which are always the same four chairs. A rigid glue line survives that; a flexible one deforms a little on each cycle and does not come back. That is the origin of the wobble that appears at month 18 and is blamed on the wood.
The third mechanism is fastener choice. A screwed bracket or a corner block with four screws is a repair technique, not a production joint: it holds the rail in position but allows rotation, and rotation under cyclic load loosens screws in 8 to 14 months. A dowelled joint at 10 by 40 mm performs better than a screw but worse than a through tenon, because the dowels carry the bending in shear across their diameter rather than across a shoulder.
The practical test for a delivered chair is a 30 second check. Put the chair on a flat surface, hold the seat with one hand and push the top of the back leg 20 mm sideways with the other. A chair with a sound joint returns without a click. A chair with a lifting shoulder clicks once and stays 2 to 3 mm out of position. Check the same four chairs the room actually uses, not a sample from the pallet, because the failure is load-history dependent.
Metal legs: taper is a wall thickness decision
A tapered steel leg is usually a square tube swaged down, and the strength question moves from section modulus to wall thickness at the small end. A 25 mm square tube with a 1.5 mm wall has a section modulus of about 1040 cubic millimetres. A 20 mm tube with a 2 mm wall has about 790, which is 24 percent less, even though the wall is thicker. Diameter wins over wall in bending, and a designer who tapers to 20 mm to look light has taken a quarter of the strength out of the leg.
The weld at the seat frame is the other half of the story. A leg welded around its full perimeter into a 25 by 25 mm frame with a 1.5 mm wall is a stiff connection; the same leg welded on two sides, or welded to a 1.0 mm frame, will crack at the weld toe within 18 months of chair dragging. Our own welding spec for tapered legs calls for a full perimeter fillet, a 2.0 mm minimum frame wall at the joint, and a gusset inside the corner where the leg carries more than 40 percent of the seat load, which is the case on any chair with a splayed front leg.
Powder coat hides a great deal and reveals one thing: cracking at the weld toe shows as a fine line of rust at month 20 in a humid room, long before the weld fails. If a buyer inspects at delivery, the place to look is the inside corner where the leg meets the frame, not the outside face that was polished.
One metal detail is worth writing into the order and is almost never written: foot plugs. A tapered tube has a thin wall at the small end and a chair dropped onto a tile floor will dent it, which makes the chair rock. A 3 mm nylon plug with a 1 mm wall spreads that impact over the full section. A dented foot on a 20 mm tube is not repairable and the leg has to be replaced, which on a welded frame means the whole base.
How to read the load test report
A chair test report is only useful if it names the test, the load and the cycles. Three numbers cover most of what a restaurant chair needs. First, a static seat and back load of 150 kg held for 60 seconds with no permanent deformation over 2 mm. Second, a dynamic fatigue test of 20,000 cycles at 100 to 120 kg on the seat, which simulates roughly three years of a busy room. Third, a side load of 300 to 400 newtons applied at seat height to the front and side, which is the load that actually breaks tapered legs.
What the report usually omits is the sample. A test on one chair from a pilot run tells you the design is sound and says nothing about production. Ask how many units were tested and whether the test piece came off the line or out of the sample room, because a hand-built sample with 8 mm tenons and a 5 mm shoulder will pass a test that the production chair fails at month 14.
The second omission is the failure mode. A report that says pass or fail is less useful than one that says where it failed at 1.6 times the test load. A leg that fails at the shoulder tells you the joint geometry is marginal even though the chair passed. A leg that fails at the taper tells you the taper is too aggressive. Either way, the report is a diagnostic, not a certificate, and a supplier who cannot describe the failure mode has probably not read their own report.
Our own test protocol for a tapered leg chair is 150 kg static, 20,000 dynamic cycles, 400 newtons side load, and a moisture cycle of 3 days at 85 percent relative humidity before the dynamic test, because a joint that passes dry and fails wet is a joint that fails in the room. The details are on the restaurant furniture quality control page, and the point of publishing them is that a buyer can ask the next supplier for the same four numbers and compare.
Eight lines for the drawing
The drawing is where a tapered leg is either specified or guessed, and eight lines cover it. One: leg section at the seat and at the floor, in millimetres, with a taper ratio no lower than 0.58. Two: where the taper starts, measured from the floor. Three: splay angle in degrees, front and back, with anything above 8 degrees requiring a stretcher. Four: tenon thickness, depth and shoulder width, at 10 mm, 35 to 40 mm and 5 mm. Five: glue type and clamping time. Six: moisture content of the leg blanks at machining, 8 to 10 percent. Seven: foot radius and glide specification. Eight: the test protocol and the sampling rate.
Two of those eight decide most failures: the shoulder and the moisture content. A 5 mm shoulder on a joint that is machined at 9 percent moisture content and assembled within 24 hours will hold for a decade in a room that swings between 40 and 75 percent relative humidity. The same geometry machined at 13 percent and assembled a week later will open at the shoulder in the first dry season, and no amount of glue will bring it back.
This is also why the leg detail belongs on the same drawing as the seat and the back, rather than on a separate component sheet. On the JP-C014 wood dining chair the leg, the seat frame and the back rail are drawn as one assembly, because the mortise angle in the leg is set by the splay of the back and a component drawing hides that relationship until the first sample is assembled and the back does not line up.
If you are specifying a chair with a tapered or splayed leg and want the joints checked before the sample is built, send your floor plan with the seat height, the cover count and the floor type and we will come back with a leg schedule rather than a catalogue page. The same applies to the open back dining chairs and to any walnut dining chairs in the scheme, where a visible open frame makes a poor joint easier to see and therefore less forgivable than a fully upholstered back.