The force that is capped on purpose

Every use of friction so far in this collection has been defensive. A bearing delivers a force nobody wanted, a wall grips soil that was supposed to push, a body slides when the check said it would not — and in each case the coefficient is a number nobody measured for the surfaces actually built, quoted to two figures and trusted to one.

There is one device in which that inequality is the whole point. A friction damper is a bolted joint with slotted holes, assembled at a specified preload on a specified surface, and it exists so that the force in whatever is behind it cannot exceed a number chosen at the drawing board.

A cap is not a capacity #

The distinction the whole subject turns on is between two things that a design calculation writes down the same way.

A capacity is a force above which something breaks. Everything is arranged so that the demand stays below it, and the consequence of exceeding it is a failure.

A cap is a force above which something moves. The demand cannot exceed it, because the device gives way and stops transmitting; the consequence of reaching it is displacement, and the structure behind it never learns that anything larger was available.

The slip load in the figure is a cap. Whatever the ground or the wind is doing, the brace holding this damper delivers 560 kN to the frame and not one newton more, until the slot runs out. That is a statement about the rest of the structure, and it is the reason the device is worth its complication: every member downstream of a cap can be designed for a force that is known exactly, rather than for a force that is the output of an analysis.

The slip load is a product of four numbers, all of them chosen #

The resistance is

$$F_{\text{slip}} = n \, \mu \, F_p \, m$$

with $n$ the bolts, $\mu$ the coefficient of the prepared surface, $F_p$ the preload in each bolt and $m$ the number of sliding interfaces. There is no area, no yield strength and no section property anywhere in it, which is the ordinary friction law doing exactly what it always does.

What is different here is that all four factors are specified rather than found. The bolts are counted, the preload is applied with a calibrated method and checked, the surface is prepared to a class, and the number of faces is a matter of how the plates are stacked.

Preload is the factor that can be adjusted after the steelwork exists, which makes it the one used for tuning. It is also the factor that does not stay where it was put: a preloaded bolt loses tension to relaxation and to creep in the packing under its head, and a device whose cap depends linearly on preload has a cap that drifts downward over years.

The coefficient is the factor with the most scatter and the least documentation, which is why real dampers rarely use steel on steel. Brass on steel, or a polymer shim, is chosen not because its coefficient is high but because it is stable: it does not change much between the first slip and the hundredth, and it does not fall away as the surfaces polish.

The same cap, reached two ways, is two devices #

Because the resistance is a product, a required cap can be met by more than one combination, and the combinations are not equivalent.

Two devices, one number. What separates them is everything the number does not contain.

The four-bolt version puts twice the load in each bolt and has half as many chances for one loose bolt to matter. Losing one bolt from four costs a quarter of the cap; losing one from eight costs an eighth.

The one-face version puts all the sliding on a single interface, so it wears twice as fast and heats twice as much per unit of energy absorbed. A friction damper turns work into heat at the contact, and the surface temperature after a long earthquake is a real design quantity — the coefficient of most sliding materials falls as they get hot.

This is the difference between a capacity check and a device specification. A capacity check is satisfied by any arrangement that reaches the number; a device has to deliver the number repeatedly, and the arrangement decides whether it can.

There is a fourth factor that does not appear in the product and decides whether any of it is real, and it is the slot. The travel available is what separates a device from a connection: a slip-resistant joint in an ordinary frame has clearance of a millimetre or two and reaches bearing almost immediately, while a damper is given twenty-five millimetres or more so that it can slide repeatedly without ever arriving there. The slot length is the design variable that turns a cap into a mechanism, and it is sized from a displacement rather than from a force — which makes it the one dimension in the device that an equilibrium calculation cannot supply.

Which free body produced the number #

Cut the damper on one of its sliding planes and take the plate on one side as the free body.

Crossing the cut are the normal forces of the bolts pressing the plates together — $n F_p$ in total — and the shear traction the contact delivers. The bolts do not cross the cut in shear, because they sit in slots and are nowhere near the ends of them; they cross it only as clamping force.

The friction bound then applies to the resultant of the shear traction, giving $\mu\,n F_p$ per interface. With the plates stacked so that the middle one slides against two outer ones, the same free body has two such surfaces and the cap doubles.

The essential point is what is not on the free body. No length of plate, no bolt diameter and no steel grade appears anywhere, which is why the two 280 kN devices above have the same drawing at the scale a calculation sees them. All three of those quantities return the moment the slot runs out and the bolts begin to bear, and the second mechanism is an ordinary bearing connection with a capacity computed the ordinary way.

What a cap does to everything behind it #

The reason to accept a device with an uncertain coefficient is that it makes something else certain.

A braced frame without a damper delivers to its beams, its columns and its foundations whatever force the brace can develop, and what the brace can develop is its own buckling or yield load — a number with its own overstrength, its own material scatter and its own strain-rate sensitivity. Sizing the members behind it means guessing high, which is the whole of capacity design: find the weakest link, then make everything else stronger than the strongest that link could turn out to be.

A cap replaces that estimate with a specification. The connection at the end of the brace slips at 560 kN, so the beam sees 560 kN, the column sees 560 kN and the foundation sees 560 kN, with no overstrength allowance because there is no strength involved. The thing that decides the force is a bolt tension rather than a steel grade, and bolt tension is the one quantity on a steel frame that is directly measured on site.

That inverts the usual relationship between a connection and its members. Ordinarily a connection is designed to the members’ capacity; here the members are designed to the connection’s cap, and the connection is the only element in the load path whose behaviour is chosen rather than inherited.

Where the cap has to sit in the load path #

A cap protects only what is behind it, and “behind” is decided by the drawing rather than by intention.

The clearest failure of that reasoning is a chevron brace. Two braces meet the underside of a beam at midspan, and when one of them reaches its limit and the other does not, the difference between them is delivered to the beam as a vertical point load nobody sized it for — the force the brace leaves behind. Putting a friction cap at the end of each brace does not remove that force; it fixes it. The unbalanced load becomes the difference between the two caps rather than the difference between a yield force and a buckling force, and it is now a number rather than an estimate, but the beam still has to carry it.

The same reading decides where the device is worth putting. A cap between a brace and a frame protects the frame. A cap between a frame and its foundation protects the foundation and leaves the frame exposed to whatever the brace can develop. A device that limits a force does nothing for anything upstream of it, and the members between the source of the load and the cap have to be designed as though the cap were not there.

There is a third position that is the most useful and the least common: a cap at the point where a structure meets something whose capacity is expensive to establish. Rock anchors, existing foundations and heritage masonry are all cases where the resistance is a survey result rather than a design, and a device that guarantees the demand can be more economical than an investigation that establishes the supply.

The loop, which is where the energy goes #

A cap that is reached once is a load case. A cap reached repeatedly is a damper, and the difference is what happens on the way back.

Each traverse of the loop dissipates its enclosed area. A device that slips at $F_{\text{slip}}$ and travels $\pm d$ removes $4 F_{\text{slip}} d$ per full cycle — a rectangle, which is the largest loop any device with that force limit can enclose, because the force is at its maximum for the whole of the travel rather than only at the ends of it.

That is a genuinely useful property and it is the reason friction dampers are efficient per unit of force. It comes with the awkward property visible in the same figure: every circuit leaves the structure displaced from where it began. There is no restoring force at all inside the slot, so a friction damper has no preference for the origin and a structure fitted with them accumulates a permanent offset. Real devices are therefore paired with something elastic — the brace itself, or a separate re-centring element — and the combination is a design about position as much as about energy.

Rate independence, and why it is unusual #

The other mechanism that removes energy from a structure is viscous, and the two are worth putting on one drawing.

A viscous device delivers a force proportional to velocity, so its loop is an ellipse, its peak force arrives at zero displacement — out of phase with the structure’s own peak — and its dissipation depends on how fast the structure is moving. A frictional device delivers a constant force, so its peak arrives with the displacement, in phase with the elastic force it is adding to.

That phase difference decides which is preferred. A viscous damper adds no force at the instant the structure is at its worst displacement, which is why it can be added to an existing frame without increasing the demand on the columns. A friction damper adds its full cap at exactly that instant, which is why the frame has to be designed for it — and why, once it has been, the frame is designed for a number rather than for an estimate.

It also means the friction device works at any speed, including very slow ones. Damping in an ordinary structure is measured rather than designed and is mostly a mystery; this is one of the few mechanisms in the subject whose contribution can be written down before the structure exists, which is the same argument that makes a deliberately weak element attractive and reaches it without any yielding at all.

What the device costs #

Nothing above is free, and the price is paid in three currencies that a force calculation does not show.

Movement. The cap works by sliding, so the structure containing it is softer than the same structure without it, at every load above the cap. That is a serviceability question rather than a strength one, and it is usually the reason a slip load is set high rather than low: a device that slips under a common wind is a device that has turned a stiff frame into a mechanism for the sake of an event that may never arrive.

Inspection. A friction damper’s condition is not visible. A bolt that has lost half its preload looks exactly like one that has not, the sliding surfaces are inside the joint, and the only external evidence of past slip is a witness mark. Every other element in a steel frame declares its state by being straight or not.

Certainty about the wrong quantity. The cap is known and the displacement at which it is reached is not, because that depends on the stiffness of everything else. A design that fixes the force has moved the uncertainty into the deformation, which is exactly the trade a displacement-based method makes deliberately — and it is a trade rather than an improvement.

Where the model stops #

The rising branch is not solved. The elastic part of every curve above is the plates shearing and the bolts bending in their clearance, and it is drawn rather than computed. The slip load and the bearing load are real; the stiffness between them is a sketch.

Static and kinetic coefficients are treated as one. They are not, and the difference produces a spike at the start of each traverse — the device breaks away at a higher force than it slides at, so the true loop has ears on it. That spike is a demand on the members behind the cap that the cap does not describe.

Nothing here is a temperature calculation. Every joule enclosed by those loops arrives at the sliding surface, and the coefficient of most damper materials falls with temperature. A long-duration event is therefore a device with a falling cap, which is safe for the frame and bad for the dissipation.

The preload is treated as constant. It is not: it falls with relaxation, with paint creep and with each slip event as the surfaces wear and the grip length shortens. A device whose cap is proportional to preload has a cap with a maintenance schedule.

And the slot is finite. Everything above assumes the travel is within the slot. Past it the device is a bearing connection, the cap is gone, and the structure behind it sees whatever the analysis said — which is exactly the condition the damper was installed to prevent, arriving without warning at the one displacement nobody plotted.

The ladder from here #

Later rungs on this anchor: the order of loading, and structures whose friction forces depend on how they were assembled rather than on what they carry. Static against kinetic coefficients as a stability question, and the stick-slip that follows from the difference. Friction in three dimensions, where the bound becomes a cone and the admissible set stops being an interval. The limit theorems for frictional systems, which are the general statement of why a problem with an inequality in it has a set of answers rather than one. And the re-centring problem, which is the price of the parallelogram and is a question about where a structure ends up rather than about what it survives.

The idea is older than its seismic application. A slotted, preloaded joint has been used to protect masts, cranes and conveyor gantries from overload for as long as there have been friction-grip bolts, and the earliest deliberate uses are in machinery rather than in buildings — a clutch is a friction damper whose slip load is chosen so that a motor can stall without breaking a shaft, and the calculation is the same product of four numbers.