A structure has more than one period

The last essay treated a whole structure as one mass on one spring, got a period out of it, and used that period to decide what counted as a sudden load. It works, and it is a lie of a particular kind: a structure has as many periods as it has ways of moving.

Five floors, five degrees of freedom, five periods. The count is exact and it is not a coincidence and it is the first thing to understand: the number of periods is the number of independent ways the masses can be arranged, which is the number of coordinates it takes to say where everything is.

Why the count is what it is #

Give each floor a horizontal position. Five floors, five numbers, and once those five are known the structure is completely described. Write the equilibrium of each floor — its mass times its acceleration, against the storey shears above and below it — and there are five equations.

$$[M]\{\ddot u\} + [K]\{u\} = 0$$

Look for a motion in which everything moves in step, so that $\{u\} = \{\phi\}\sin\omega t$ and every floor reaches its extreme at the same instant. Substituting gives

$$([K] - \omega^2[M])\{\phi\} = 0$$

which has a non-trivial solution only when the matrix is singular — and there are exactly five values of $\omega^2$ that make it so, because the determinant is a fifth-degree polynomial. Each gives a shape $\{\phi\}$.

That is the whole derivation, and it is worth noticing what kind of argument it is. The condition for a mode to exist is that a set of equations be singular — the same condition that makes a truss a mechanism rather than a structure, and the same determinant that decides whether a frame has a buckling load. Modes, mechanisms and buckling modes are the same mathematics asked three times, which is why the fleet’s word for all three is critical form.

They are not harmonics #

The spacing of a structure’s periods is a property of how its mass and stiffness are distributed, and there is no general law about it. What there is instead is a theorem about the shapes, and it is exact: the nth mode crosses the axis n−1 times.

That is a Sturm–Liouville result rather than a drawing convention, and nothing in the eigensolver knows about it — which makes it a genuine check on the arithmetic rather than a restatement of it. The first mode has every floor moving the same way; the second has one storey level where nothing moves at all; the third has two. Counting the crossings in the figure identifies which mode is which, without reference to any frequency.

That closed form is worth having for a reason beyond checking. It says the frequencies of a uniform frame are sines of equally spaced angles, which crowd together as the angle approaches a right angle — so the periods of a tall uniform building bunch up at the short end, and a frame with twenty storeys has its highest four modes within a few per cent of each other. Structures with closely spaced modes behave badly under any analysis that treats the modes separately, and this is where they come from.

What a mode is not #

Three misreadings are common enough to be worth naming, and all three come from the pictures.

A mode shape is not a deflection. It has no units and no size. The figures above are drawn to some convenient amplitude, and doubling every number in a mode shape gives the same mode. What is fixed is the ratio between the floors — the top floor moves 1.83 times as far as the second in the first mode of the five-storey frame, and that ratio is the content.

A mode is not a load case. No load produces “the second mode”. A load produces a response, and the response can be decomposed into modes, usually with the first dominating. Asking what force causes the second mode is asking the question backwards.

A structure is not in one mode at a time. At any instant it is in a superposition of all of them. What makes the decomposition useful is not that only one is present but that they can be computed separately and added, which is orthogonality again.

Orthogonality, which is what makes any of this usable #

Here is the property the whole of structural dynamics is built on, and it is not obvious.

Take any two distinct modes, multiply them floor by floor, weight each product by that floor’s mass, and add:

$$\{\phi_i\}^{\mathsf T}[M]\{\phi_j\} = 0 \quad (i \neq j)$$

Zero. Every time, to machine precision, for every pair. And the same holds through the stiffness matrix.

What that buys is worth stating slowly. It means the modes are independent coordinates. A motion of the building — any motion, however complicated — can be written as a sum of modal contributions, and when it is, the equations of motion separate: each mode’s amplitude obeys its own single-degree-of-freedom equation, with its own period, its own damping, and its own share of the applied load, and no term in it involves any other mode.

So an eight-storey building under an earthquake is not an eight-degree-of-freedom problem to be solved as one. It is eight separate problems of the kind the last three essays solved, each with one mass, one spring and one dashpot, and their answers added at the end. Every result about suddenly applied loads, about resonance and about damping transfers intact.

That is why this field can be taught with a mass on a spring at all. Not because structures are simple, but because the mathematics decomposes.

What breaks it #

Two things, and they are the two places where computational dynamics gets expensive.

Damping does not generally decompose. The mass and stiffness matrices are orthogonalised by the mode shapes; the damping matrix, whatever it is, need not be. The universal fix is to assume it is — to assign each mode a damping ratio directly, which is Rayleigh damping or modal damping — and it is a fix rather than a derivation. Given how little is known about where damping comes from, assuming a convenient form for it is not the largest approximation in the calculation.

Nonlinearity destroys it completely. Once a structure yields, the stiffness matrix changes as the response develops, so the modes change while the motion is happening. There is no meaningful mode shape for a frame with plastic hinges in some storeys and not others, which is why inelastic seismic analysis is done by integrating the whole system in time rather than mode by mode.

The second mode is where the surprises live #

Read the second mode’s shape and something practical falls out. In the first mode the largest drift — the difference in displacement between adjacent floors, which is what damages cladding and columns — is at the bottom, where the shape rises steeply. In the second mode the shape reverses in the middle, so the drift peaks near mid-height and the top floor moves in the opposite direction to the ground floor at the same instant.

That matters most for the things attached to the building rather than for the building. A pipe run, a lift guide rail or a cladding rail spanning several floors is stretched by the difference between floor motions, and a second mode does that where a first mode does not.

The frame is a caricature, on purpose #

Everything above uses a shear building: mass concentrated at the floors, stiffness in the columns between them, floors that do not rotate. Real frames do not behave that way, and the model is kept because it is the smallest structure with more than one period.

Where the caricature misleads:

Real floors rotate. A frame with flexible beams deforms with the joints turning, which lowers every frequency and changes every shape. The model above effectively assumes infinitely stiff beams, which is the rigid-joint assumption the connections field spends its time undoing.

Real buildings twist. A plan without symmetry has modes that are part sway and part rotation, and a torsional mode close in period to a translational one produces corner displacements that neither mode predicts alone. That coupling is the commonest reason a building performs worse than its analysis suggested.

Real masses are not at the floors. They are, mostly, which is the one part of the model that survives contact with reality — a building’s mass genuinely is concentrated in its slabs.

Where the mass is, mode by mode #

This is the quantity that makes the mode count manageable, and it is the next rung’s subject. Eight modes exist; two of them carry 95% of the mass; and for a great many purposes the rest can be ignored — provided the purpose is a force. For an acceleration it cannot, which is the interesting half.

A soft storey, seen as an eigenvalue #

The concentration is the danger and the mass ratio is the tell. A structure whose first mode carries a larger share of the mass is one whose deformation has become concentrated, and concentration of deformation is concentration of damage. The same arithmetic that made the building simpler to analyse made it worse to be in.

That stability is worth a sentence of its own, because it is the reason the field has rules of thumb at all. The mass distribution across modes barely moves as a building gets taller: the first mode always takes most of it — 91.4% at three storeys, 85.6% at eight, 84.2% at twelve — the second always takes about a tenth, and the tail is always negligible. What changes with height is the period, which grows roughly in proportion to the number of storeys — so a tall building is not a short building with more modes to worry about, it is a short building with the same modes at slower rates.

Why closely spaced modes behave badly #

The bunching noticed above — a uniform tall frame’s higher periods crowding together — was left as an observation. It has a consequence, and it is the one place where the orthogonality that makes this field tractable stops paying.

Orthogonality separates the equations. Each mode is solved alone and produces its own answer, and the answers have to be put back together. The difficulty is that each mode reaches its peak at a different instant, so adding the peaks assumes a coincidence that does not happen and gives an answer that is much too large.

The standard remedy is to combine them as a square root of the sum of squares:

$$R \approx \sqrt{\textstyle\sum_n R_n^2}$$

and it is worth being clear what kind of statement that is. It is not a mechanical result; it is a statistical one, correct if the modal peaks behave as independent random quantities. Two modes at 1 Hz and 3 Hz respond to different parts of an irregular input, so their peaks arrive at unrelated moments and the assumption holds well.

Two modes at 1.00 and 1.05 Hz do not. They are driven by the same part of the input, they respond nearly in step, and their peaks arrive together — so the true combination is close to the plain sum. For two equal modal responses that is $2a$, and the root-sum-square gives $1.414a$: the standard rule returns 71 per cent of the answer, and it errs in the unsafe direction, which is the one direction a combination rule is not allowed to err in.

The fix is the complete quadratic combination, which adds the cross terms back with a correlation coefficient that depends on how close the two frequencies are and on the damping — near one for a pair almost coincident, near zero for a pair well separated, so it reduces to the root-sum-square when the modes are apart and to the plain sum when they are together.

Where the closely spaced pairs come from is the part worth carrying, because it is not where an intuition would look.

A plan-symmetric building. Symmetry gives the two horizontal directions nearly identical stiffness, so the two fundamental translational modes are at nearly the same period by construction. They are orthogonal in direction and adjacent in frequency, which is precisely the condition the root-sum-square handles worst.

A building whose torsional period is near a translational one. This is the arrangement that produces the worst corner displacements, and it is also the arrangement where the combination rule is least reliable — the two errors compound.

And any uniform tall frame at its upper modes, by the sine formula this essay derived: the frequencies crowd toward the top end of the range, so the higher modes of a twenty-storey building are within a few per cent of one another.

There is one more source and it is the one that catches retrofits. Adding a device tunes two modes onto one another deliberately — that is what a tuned mass damper does, and the whole of its benefit comes from the two resulting modes being close together and strongly coupled. A structure with one fitted has been given, on purpose, exactly the configuration that a modal combination rule handles worst, which is why such systems are checked by integrating the response in time rather than by combining modal peaks at all.

Which gives an inversion worth stating plainly. The building most likely to need a careful combination rule is the regular, symmetric, uniform one — the building that looks least likely to have a dynamics problem, and the one a designer is least likely to interrogate. Irregularity separates the periods and makes the modes independent; symmetry brings them together and makes them correlated, and the analysis that treats them as separate answers is quietly assuming the opposite of what the geometry provides.

Modes are measured, not only computed #

Everything above is arithmetic on a model. The modes of a real building can be measured, and the measurement is worth describing because it is how the models get checked.

Put accelerometers on several floors and record for an hour with nobody doing anything special — traffic, wind and lifts provide the excitation. Every mode is being nudged constantly by that noise, and each responds most at its own frequency, so the recorded spectrum has a peak at each period. The shape comes from the relative amplitudes and signs at the different floors at that frequency: floors moving together at 0.93 Hz and in opposition at 3.2 Hz identify the first and second modes without any force ever being measured.

Two things usually come out of such a test and both are humbling. The measured periods are commonly shorter than the computed ones — real buildings are stiffer than their models, because the models leave out cladding, partitions and everything else that was not designed to carry load. And the measured mode shapes are rarely as clean as the drawn ones, because real buildings twist as they sway.

The first of those has a consequence worth carrying: a model that is calibrated to a measured period is a model that has had its stiffness corrected, and its strength predictions were never wrong in the first place. Dynamics is the one branch of this subject where the analysis can be checked against the finished structure without breaking it, and that fact is what makes the field’s assumptions unusually well tested.

Where the ladder goes #

Three directions.

The first is the practical one: given that a structure has many modes, how many have to be considered? The answer is a mass count rather than a period count, and it is the next essay.

The second is what happens when the excitation is not a force but the ground moving, in which case each mode is driven in proportion to how much mass it moves — and the spectrum is the tool that gives each mode its answer.

The third is what a second mass on a second spring can be made to do deliberately. It is the only place in this subject where adding a degree of freedom is the design rather than the difficulty. Hang one from the top of a structure, tune it, and the single mode it was added to becomes two modes that are both smaller — the one place in this field where an extra degree of freedom is the answer rather than the problem.