Structural Engineering & Statics
where the load goes
Dynamics

The force that belongs to the model

When two buildings meet, momentum decides what each of them feels, and no contact law can change it. What the contact law decides is the force, and the force it reports is the contact stiffness somebody assumed, raised to a power. Four laws and a hundredfold change of stiffness move the buildings by a few per cent and the force by a factor of eight.

Assumes The gap between two buildings, The block that is safer for being bigger and Twice the deflection, for the same load.

14 min read 7 figures The load that will not hold stillWhich failure arrives firstThe load must go somewhere

Two buildings too close together will meet in an earthquake, and the gap between them is sized so that they do not. When they do, the first thing momentum says is exact. A 500 tonne building and a 300 tonne building closing at 1.94 m/s, with a coefficient of restitution of 0.65, exchange an impulse of 600 kN·s, and the lighter building’s velocity changes by more, in proportion to the masses.

The floor that arrives at a column then said the uncomfortable thing: the force behind that impulse cannot be computed to better than an order of magnitude, because it depends on how long the contact lasts, and that depends on the stiffness of two concrete faces meeting. Design proceeds by avoiding the arrangement rather than checking it.

An analysis of pounding does compute a force, though, and it does so by putting a contact element in the gap. There are four in common use. This essay puts all four on one collision and then on one earthquake, and separates what each answer owes to the buildings from what it owes to the element.

One collision, four contact laws, one impulse. Contact force against time for one collision between 500 t and 300 t closing at 1.94 m/s, under four contact laws sharing one contact constant of 2.8×10^9 N/m^1.5. The elastic linear spring peaks at 19.7 MN and lasts 58 ms; the Hertz law peaks at 22.0 MN over 61 ms. Set to a restitution of 0.65, the spring and dashpot peaks at 16.8 MN and ends pulling at 3.5 MN, a tension two faces in contact cannot carry, while the damped Hertz law peaks at 20.1 MN and returns a restitution of 0.77 instead. The areas under the curves are the impulses: 600 kN·s for the spring and dashpot, which is what momentum requires at 0.65, 646 for the damped Hertz law, and 727 for both elastic laws.
Fig. 1 Contact force against time for one collision between 500 t and 300 t closing at 1.94 m/s, under four laws sharing one contact constant. The linear spring peaks at 19.7 MN over 58 ms and the Hertz law at 22.0 MN over 61 ms. The spring and dashpot peaks at 16.8 MN and ends pulling at 3.5 MN; the damped Hertz law peaks at 20.1 MN and returns a restitution of 0.77 rather than 0.65.

Momentum sets the impulse, and the law only shapes it

The four laws are the choices an analyst actually has. A linear spring pushes back in proportion to how far the two faces have pressed into each other. A spring and dashpot, usually called a Kelvin element, adds a force in proportion to how fast they are pressing. The Hertz law pushes back as the indentation to the power one and a half, which is what two elastic curved bodies do as their contact area grows. The damped Hertz law multiplies that by a factor that grows with the closing speed, so that it too can lose energy.

To compare them fairly they are given one contact. The Hertz constant is fixed, and the linear spring takes the stiffness the Hertz law has at its own deepest indentation. The two damped laws are set to the same restitution of 0.65.

The four force histories differ in height, in length and in shape. The areas under them do not differ in any way that is not forced. Both elastic laws enclose 727 kN·s, which is exactly twice the effective mass times the closing speed: a perfectly elastic impact reverses the relative velocity. The spring and dashpot encloses 600 kN·s, which is the same quantity with 1.65 in place of 2, exactly what a restitution of 0.65 requires. The impulse is a statement about momentum, and every law must deliver the impulse its restitution asks for. Nothing about the contact’s stiffness enters it.

That is the first separation, and it is the useful one. Each building’s velocity change is the impulse divided by its mass: for a restitution of 0.65, 1.20 m/s for the heavier and 2.00 m/s for the lighter, whatever element delivers it. The motion of the buildings after the collision is decided before the element is chosen.

Decided, that is, by the restitution, which is the one parameter every law needs right. For a block landing on its own base the restitution follows from the block’s proportions, and even there it is a number nobody measures. For two buildings it follows from nothing that can be drawn: it is how much of the closing energy goes into crushing a slab edge, cracking cover and bending cladding, and it changes from one collision to the next as the edge is damaged. The impulse depends on it only through one plus the restitution, so its uncertainty is bounded — a restitution anywhere from 0.5 to 0.8 moves the impulse by less than a tenth either way. That is small beside what the contact stiffness does to the force.

Where each law puts the energy the collision loses

Where each law puts the energy the collision loses. Contact force against indentation for one collision between 500 t and 300 t closing at 1.94 m/s, under four contact laws sharing one contact constant of 2.8×10^9 N/m^1.5. The two elastic laws load and unload along one curve, a straight line for the spring and a curve steepening as the 1.5 power for Hertz, and lose nothing. The spring and dashpot jumps to 5.4 MN the instant the faces touch, because its dashpot resists the full closing speed at zero indentation, and it leaves at 3.5 MN of tension; its loop holds 58 per cent of the kinetic energy of the approach. The damped Hertz law starts and ends at zero force and loses 40 per cent.
Fig. 2 Contact force against indentation for the same collision. The elastic laws load and unload along one curve and lose nothing. The spring and dashpot jumps to 5.4 MN the instant the faces touch and leaves in tension, losing 58 per cent of the approach energy; the damped Hertz law starts and ends at zero and loses 40 per cent.

Drawn against indentation rather than time, each law’s loss is the area its loop encloses, and the two damped laws put it in very different places.

The spring and dashpot’s dashpot resists velocity, and the velocity is largest at the instant of first touch, when the indentation is still zero. So the force jumps from nothing to 5.4 MN in no time at all, the dashpot pushing against the full 1.94 m/s before the spring has been compressed by anything. At the other end of the contact the faces are separating at speed, the dashpot resists that too, and the force is negative when the indentation returns to zero. The loop is lopsided, with a vertical edge at both ends, and it encloses 58 per cent of the approach energy, which is exactly what 1−e21 - e^2 gives for a restitution of 0.65.

The damped Hertz law’s extra force is proportional to both the velocity and the indentation, so it is zero at first touch and zero at separation. Its loop is smooth and closes at the origin. It encloses only 40 per cent of the energy, though, which is a symptom of the second artefact below.

The force is the stiffness somebody assumed

The force is the stiffness somebody assumed. Peak contact force against the contact constant, from 1.0×10^8 N/m^1.5 to 1.0×10^12 N/m^1.5, for the same collision between 500 t and 300 t at 1.94 m/s under all four laws. Over four decades of stiffness the Hertz force rises from 5.9 MN to 233.0 MN, as the constant to the power 0.4, and the contact shortens from 229 ms to 6 ms. The impulse does not move at all: 727 kN·s for the elastic laws at every stiffness, because momentum fixes it. At the 2.8×10^9 N/m^1.5 marked the Hertz force is 22.0 MN.
Fig. 3 Peak contact force against the contact constant, over four decades, for the same collision under all four laws. The Hertz force rises from 5.9 to 233 MN, as the constant to the power 0.4, and the contact shortens from 229 ms to 6 ms. The impulse is 727 kN·s for the elastic laws at every stiffness.

The impulse was fixed. The force is the impulse spread over the contact duration, so everything the element decides is in the duration, and the duration is decided by the stiffness.

For a linear spring the arithmetic is the one a dropped weight obeys: the peak force is the closing speed times the square root of the stiffness times the mass, so a hundred times the stiffness is ten times the force. For the Hertz law the exponent is different. The deepest indentation goes as the contact constant to the power minus two fifths, the peak force as the constant to the power two fifths, and so a hundred times the constant is 6.3 times the force. Over the four decades drawn, the force on one collision runs from 5.9 MN to 233 MN, and the contact from a quarter of a second to six milliseconds.

Nobody knows the contact constant to within that range. It depends on the curvature of two faces that are not curved in any designed way, on the modulus of cover concrete and cladding, and on whatever was left in the joint. Published values span orders of magnitude, and the essay on floor-to-column strikes was right that the force is not calculable. The contact law does not change that, whichever law it is. It takes an assumed stiffness and returns the force that stiffness implies, to three significant figures.

That is the reason the soft layer in a joint matters more than anything behind it. In a chain of stiffnesses, the softest one in series takes most of the deformation and sets the force for the rest, and a crushable filler in a seismic gap is a deliberate choice of the softest spring. It is the one way to make the contact stiffness a designed number instead of an assumed one.

A dashpot that pulls

The spring and dashpot has one property nothing else in this problem has. In the first collision its force at separation was minus 3.5 MN: the element was holding the two buildings together as they tried to part.

Two faces in contact can push and cannot pull. A contact that ends in tension is a contact element doing something the buildings cannot, and the tension is not small. It is a fifth of the peak force here, and it grows with the stiffness in proportion, because a stiffer spring gives a faster separation and the dashpot’s force follows the speed. The usual correction is to clip the element’s force at zero. That stops the tension, but the dashpot’s work during the pulling phase was part of the energy the element was set to lose, so a clipped element loses less and returns a higher restitution than it was given. The spring and dashpot is not the more accurate of the two damped laws; it is the one whose error is easier to see.

The damped Hertz law misses its own restitution

The restitution a law is set to, and the one it returns. The restitution each damped law actually returns from one collision, against the restitution its damping was calculated for, from 0.30 to one. The spring and dashpot lies on the diagonal, because its dashpot is derived from the exact solution of a damped linear oscillator. The damped Hertz law lies above it everywhere below one: set to 0.65 it returns 0.775, and set to 0.30 it returns 0.684, because its damping term is derived for impacts that lose little energy and concrete faces lose a great deal.
Fig. 4 The restitution each damped law returns from one collision, against the one its damping was set for. The spring and dashpot lies on the diagonal. The damped Hertz law lies above it: set to 0.65 it returns 0.775, and set to 0.30 it returns 0.684.

The damped Hertz law avoids the tension, and pays for it elsewhere. Its damping coefficient comes from an energy balance that assumes the impact loses little of its energy, which is true of steel balls and false of concrete edges. Set to a restitution of 0.65, it returns 0.775. Set to 0.30, it returns 0.684. Below about 0.9 it cannot be made to lose as much energy as it was asked to, and the impulse it delivers is 646 kN·s instead of 600.

The direction of that error is worth noticing. A larger restitution is a larger impulse and a larger velocity change for each building, so the damped Hertz law, with its coefficient taken from the formula, overstates the collision’s effect on the buildings. Corrected damping coefficients exist for exactly this reason, fitted for low restitutions, and a pounding analysis that uses the original formula is analysing a harder collision than its input says.

Two buildings through a whole pulse

A single collision is idealised: two free masses closing at a given speed. In an earthquake the two are buildings with their own stiffness and damping, the closing speed comes out of the analysis, and they meet more than once.

Two buildings, one pulse, and the collisions between them. The displacements of a 500 t building with a 0.8 s period and the face of a 300 t building with a 1.2 s period drawn 50 mm from it, under one 1.0 s sine pulse of 0.50 g, with the Hertz contact law at 2.8×10^9 N/m^1.5. They meet seven times; the hardest impact is at 1.44 s, closing at 2.85 m/s, lasting 56 ms and peaking at 35.0 MN. The first building reaches 244 mm against 220 mm on its own, dashed, and the second 410 mm against 419 mm.
Fig. 5 The 500 t building with a 0.8 s period and the face of the 300 t building with a 1.2 s period, 50 mm away, under one 1.0 s sine pulse of 0.5 g with the Hertz law. They meet seven times; the hardest impact is at 1.44 s, closing at 2.85 m/s, lasting 56 ms and peaking at 35.0 MN. The first building reaches 244 mm against 220 mm on its own; the second 410 mm against 419 mm.

The pulse drives the two buildings out of phase almost at once, because their periods differ, and the stiffer one runs into the softer seven times. The hardest impact is not the first. It comes at 1.44 s, after the pulse has ended, with the two closing at 2.85 m/s. The quick estimate of a closing speed takes each building’s largest displacement on its own, multiplies it by its frequency and adds the two, which for this pair gives 3.92 m/s. The true closing speed is well below that for the reason the gap is not the sum of the displacements: the two buildings are not each at their fastest at the moment they meet.

The collisions change the buildings’ motion in a way that is easy to read off the figure. The stiffer building is pushed back each time and swings further the other way, reaching 244 mm against the 220 mm it reaches on its own. The softer building is stopped short of its free swing and reaches 410 mm instead of 419. Neither number is dramatic. Both belong to the buildings rather than to the element, as the next two figures show.

A stiffer contact, and a tension nobody built

Two buildings, one pulse, and the collisions between them. The displacements of a 500 t building with a 0.8 s period and the face of a 300 t building with a 1.2 s period drawn 50 mm from it, under one 1.0 s sine pulse of 0.50 g, with the spring and dashpot contact law at 2.8×10^11 N/m^1.5. They meet seven times; the hardest impact is at 1.44 s, closing at 2.86 m/s, lasting 9 ms and peaking at 169.4 MN. The first building reaches 233 mm against 220 mm on its own, dashed, and the second 420 mm against 419 mm. Its contact ends in tension, pulling at up to 34.1 MN.
Fig. 6 The same pair and pulse with a spring and dashpot a hundred times stiffer. They meet seven times; the hardest impact is at 1.44 s, closing at 2.86 m/s, lasting 9 ms and peaking at 169.4 MN, and the element pulls at up to 34.1 MN. The first building reaches 233 mm and the second 420 mm.

A hundredfold stiffer contact and the other damped law change the analysis a great deal where the element lives. The hardest impact is at the same moment, at essentially the same closing speed, and it now lasts 9 ms and peaks at 169 MN — nearly five times the force of the Hertz analysis. The element’s tension has grown with it, to 34 MN pulling the buildings together at every separation.

And the buildings barely moved differently. The first reaches 233 mm, the second 420 mm. The impact happened at the same time, the same number of times, with the same momentum exchanged, and the buildings responded to the momentum.

What the buildings feel does not depend on the law

What the contact law changes, and what it does not. The largest displacement of each building and the largest contact force, for the same two buildings and pulse under all four contact laws, each at a contact constant of 2.8×10^9 N/m^1.5 and a hundred times that. Across all eight analyses the first building's largest displacement lies between 226 and 244 mm and the second's between 400 and 437 mm, against 220 and 419 mm with no contact. The largest contact force lies between 27.3 MN and 225.5 MN, a factor of 8.3.
Fig. 7 The largest displacement of each building and the largest contact force, for all four laws at the reference contact constant and a hundred times it. The first building lies between 226 and 244 mm and the second between 400 and 437 mm, against 220 and 419 mm with no contact. The largest contact force lies between 27.3 and 225.5 MN, a factor of 8.3.

Across all eight analyses, the first building’s largest displacement moves by 8 per cent from the smallest to the largest, and the second building’s by 9. The largest contact force moves by a factor of 8.3.

That split is the answer to the question this essay started with. The displacements a pounding analysis reports belong to the buildings, and the forces it reports belong to the model. A displacement is an integral of the momentum exchanged, and momentum is fixed by the masses, the closing speed and the restitution. A force is that momentum divided by a duration, and the duration is whatever the assumed stiffness makes it. Change the contact element and the drift check barely moves, while the local force check moves by close to an order of magnitude — the uncertainty in the force, measured rather than estimated.

It is also why a pounding analysis keeps the character of the displacements it starts from. The displacement a design spectrum gives is the peak of a response rather than a load, and the correction pounding makes to it is a correction to a response: modest, and governed by the momentum the buildings exchange rather than by anything local to the faces that meet.

It follows that the two outputs deserve different treatment. The displacement and drift results of a pounding analysis can be used with ordinary confidence, whichever element produced them, provided its restitution is right. The contact forces cannot be used as design loads at all without a stated range, and the range is the range of the contact stiffness, taken to the power of the law.

Choosing a law for what it is being asked

Once the separation is clear, the choice of element stops being a question of which law is best and becomes a question of which error matters for the output wanted.

For drifts and the distance the buildings travel, any of the four will do, and the restitution is the only parameter that must be right. The spring and dashpot is the easiest to set correctly, since it returns exactly the restitution it is given.

For the velocity each building is left with, which is what drives the damage in the lighter one, the restitution again decides, and the damped Hertz law with its original coefficient overstates it by returning 0.775 for 0.65.

For local forces, no law will do on its own. Every result must be repeated at the lower and upper ends of the plausible contact stiffness, and the force reported as that range. A contact detail designed against the upper value, or softened with a filler until its stiffness is known, is the only route to a force that is a design quantity. For yielding buildings the displacements carry the design anyway, which is why this separation leaves the ordinary seismic check intact.

What the two-mass model leaves out

Each building is a single mass on a spring. A real building meets its neighbour at floor levels over its height, and it has more than one period, so a collision at mid-height excites modes the single mass does not have. And where it meets depends on its floor levels rather than on its roof.

The buildings’ damping is assumed. Each has 5 per cent of critical, the value every dynamic calculation assumes because nobody designs it, and between collisions it decides how much of each swing survives to the next impact.

The buildings stay elastic. A building that has yielded has a longer period and different damping, and a collision that crushes a slab edge loses energy the restitution does not describe.

The contact is a point on one line. Real faces meet along an edge or a slab, at an angle, and slide as they meet, and the friction of that sliding is another loss.

The restitution is one number. Measured restitutions for concrete fall with closing speed, so a hard impact loses a larger share of its energy than a soft one, and a constant restitution overstates the hardest collisions.

Still open: whether a filler makes the force a design quantity

Everything above treats the contact stiffness as unknowable, and for two bare concrete faces it is. A crushable filler in the gap changes that, because it is a material with a measured force against compression and it is far softer than the concrete behind it, so it sets the contact duration by itself. Whether a filler can be chosen stiff enough not to eat the gap in ordinary movement and soft enough to lengthen a collision to tens of milliseconds, whether it survives the first impact to act on the seventh, and whether the force it delivers can then be stated as a single number rather than a range of orders of magnitude, is a question about a material nobody currently designs to that purpose.

Where this goes next

Buildings that sway alike need almost no gap between them. The separation two adjacent buildings need, against the ratio of their periods. The obvious answer is the sum of what each can do — 340 mm — and it is wrong, because the two peaks do not occur at the same instant. The right combination is the one modal responses use, √(u₁² + u₂² − 2ρu₁u₂), with ρ the cross-correlation coefficient of the two responses. ρ depends on the period ratio and behaves the opposite way to intuition: at a ratio of one the two buildings sway together, ρ = 1, and the gap collapses to the difference of the two, 100 mm. At the 0.57 drawn ρ is 0.029 and the gap is 248 mm — 27 per cent less than the sum, and 99 per cent of the square root of the sum of squares. The gap between two buildings Two towers side by side in an earthquake need a gap. The obvious answer is the sum of what each can move, and it is wrong — because the two peaks do not happen at the same instant. What decides the answer is the ratio of the two periods, and buildings that sway alike need almost no gap at all. Where two buildings touch, and what is there when they do. Closure between an 8-storey building 24 m tall and a 4-storey one 15 m tall, drawn against height, with the 50 mm gap between them. The two swaying out of phase close on each other more the higher up they are, so they first touch at 7.6 m and are in contact above it. The dots are the taller building's floors: its storeys are 3.00 m and the other's are 3.75, so two of them arrive part way up a column rather than at a slab. A blow at mid-height of a column asks it for a shear of half the impact and an end moment of Fh/8, neither of which is a demand any part of the design contains. The floor that arrives at a column The gap between two buildings is computed from their roof displacements, which is where each of them moves most. It is not where they touch, and it is not what is there when they do — a slab edge meeting a column part way up its height is a different event from two slabs meeting, and it is the one that appears in the photographs. What a drop height is worth, as a factor on the answer for a weight placed slowly. The peak displacement as a multiple of the static deflection, against the height a weight is dropped from divided by the deflection that weight causes when it is placed. The curve is 1 + √(1 + 2h/δ), which is conservation of energy and nothing else: the weight does work over the height it falls PLUS the distance the structure then gives, and the structure stores work only over the second. At a ratio of 40 the factor is 10.00, and at zero it is exactly 2 — the marked point, where a dropped weight becomes a placed one. The weight that was dropped A half-tonne load lowered onto a beam produces 5 kN. The same load dropped one metre onto the same beam produces 160 kN — and onto a beam ten times softer, 54 kN. The stiff structure is the one that suffers, which is the opposite of nearly every other rule about structures. A joint is springs in series. The five components of an end-plate joint, with each bar the flexibility it contributes. The column flange in bending is 39.62% of the total on its own, and doubling its stiffness raises the joint's by a factor of 1.25 — while doubling the stiffest component buys 1.05. The joint's rotational stiffness is 25227.71 kN·m per radian. A joint made of springs in series A connection's stiffness is not a property of the connection. It is the series combination of the flexibilities of everything the force passes through — so the softest component decides, and stiffening any of the others changes almost nothing.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

DampingEnergy dissipationImpact factorImpulseNon-linear responsePoundingRestitution