A measure of twist that divides by the twist

The corner that moves most located a plan’s centre of rigidity, found that a force applied anywhere else twists the floor, and measured the twist by how much further the far wall moves than it would under the same force with no torsion — 1.98 times for its core-and-façade plan, twelve times for a plan whose only stiffness is a central core. It ended by naming the measure the codes actually use: “torsional irregularity as codes define it — the ratio of maximum to average storey drift — and why that particular measure was chosen.”

This essay puts the two measures side by side. They are not the same number, and the difference between them is not a detail of definition.

Two readings of one movement #

The floor is rigid in its own plane, so under a sideways storey force it translates and rotates as a single body, and every point’s displacement lies on a straight line across the plan. The plan here is the earlier essay’s, 30 by 18 m, pushed across its 30 m length. It is described by two numbers: its natural eccentricity $e_n$, how far its centre of rigidity sits from the centre of its mass, and its torsional radius $r$, the walls’ torsional stiffness about that centre divided by their translational stiffness, square-rooted. The force is applied at the centre of mass shifted by the codes’ accidental eccentricity of 5 per cent of the length, which makes the lever arm $e = e_n + 0.05L_x$.

If $u_0$ is how far a plan with the same walls but no eccentricity would translate, the floor rotates by $u_0 e/r^2$ and a point a distance $x$ from the centre of rigidity moves

$$\delta(x) = u_0\left(1 + \frac{e\,x}{r^2}\right)$$

Two numbers can be read from that line. One is how far the worst edge moves compared with a plan with no torsion, $\delta_{max}/u_0$: the amplification, the earlier essay’s measure. The other is how far the worst edge moves compared with the average of the two edges, $\delta_{max}/\delta_{avg}$: the ratio that ASCE 7’s table of plan irregularities uses, a building being torsionally irregular above 1.2 and extremely so above 1.4.

For the core-and-façade plan the two readings are 1.98 and 1.63. The far edge moves nearly twice what a torsion-free plan would; the near edge, beside the core, moves less than half; and their average is 1.21 $u_0$, not $u_0$. The ratio is 1.98 over 1.21. The average the ratio divides by has already been moved by the twist.

For a symmetric plan — no natural eccentricity, the twist coming only from the accidental 5 per cent — the average is $u_0$ exactly, because the two edges move equally either side of it, and the ratio and the amplification are the same number. A symmetric plan with a torsional radius of 6.0 m reads 1.63, like the core-and-façade plan, and its corner moves 1.63 $u_0$. The same ratio; a corner that moves 21 per cent further in the eccentric plan.

Why the two part company #

The quotient of the two readings is the average itself: $\delta_{avg}/u_0 = 1 + e\,e_n/r^2$, which is one for a symmetric plan and grows with the natural eccentricity. So

$$\frac{\delta_{max}}{\delta_{avg}} = \frac{1 + e\,a/r^2}{1 + e\,e_n/r^2}$$

with $a = L_x/2 + e_n$ the distance from the centre of rigidity to the far edge. The numerator is the amplification. The denominator is what the twist does to the middle of the plan: an eccentric plan’s centre of mass is off its centre of rigidity, so when the floor rotates, the middle of the plan swings out with the far edge. The ratio divides the far edge’s movement by the middle’s, and as the eccentricity grows both are dominated by the same rotation. For a large eccentricity the ratio tends to $a/e_n = (L_x/2 + e_n)/e_n$, the ratio of their distances from the centre of rigidity, however large the rotation is.

The two curves rise together. The worst edge rises faster at first, which is when the ratio grows; then the average catches up in proportion, and the ratio between them — the codes’ measure — stops growing.

The ratio stops, the twist does not #

At this torsional radius the ratio climbs through 1.2 and 1.4 quickly, reaches 1.76 when the centre of rigidity is 0.30 of the length off-centre, and then turns down. Over the same range the corner’s movement goes from 1.2 to 4.8 times a torsion-free plan’s, and rises the whole way. Past an eccentricity of about a third of the length, moving the stiffness further off-centre makes the corner move more and the plan read more regular.

That is not a flaw that bites often at the thresholds, which the ratio crosses early. A plan with this torsional radius is flagged as extremely irregular at an eccentricity of about a twentieth of its length and stays flagged. What the ratio cannot do is rank. Two plans both flagged — one whose corner moves twice a torsion-free plan’s, one whose corner moves more than four times as far — read 1.63 and 1.74, and anything the codes compute from the ratio treats them as nearly the same building.

Where the ratio reads regular and the floor twists #

The map shows the whole family at once. For a symmetric plan — the left edge — the ratio’s contours and the movement’s are the same curves, as they must be. Moving right, they separate. The movement’s contours fall steadily: the more eccentric the plan, the more torsional radius it needs to keep its corner inside a given multiple of the translation. The ratio’s contours curve back up: at large eccentricities the ratio is set by the geometry of the plan, not by how stiff it is in torsion, and a plan with a large torsional radius and a large eccentricity reads the same ratio as one with a modest radius and a modest eccentricity while its corner moves much further.

The European code asks the question differently, and its boundaries on the map are straight. EN 1998-1 calls a plan regular if its eccentricity is at most three tenths of its torsional radius and its torsional radius is at least the floor’s own radius of gyration, $l_s$. Both tests are monotonic in the right direction: more eccentricity or less torsional radius can only fail them. The core-and-façade plan passes the second (10.6 m against 10.1) and fails the first (4.2 m against 3.2). The European tests measure the plan’s causes, eccentricity and torsional radius; the American ratio measures an effect, and an effect normalised by itself.

What the walls are asked for #

The difference is not academic, because the walls are sized by displacement. On a rigid floor each wall’s force is its stiffness times the floor’s movement where the wall stands, so a wall at the far edge carries its direct share of the storey force multiplied by exactly the amplification: the far façade of the core-and-façade plan is asked for 1.98 times its share, as the earlier essay found. Nothing in the ratio says that. A designer told the plan is “irregular, ratio 1.63” has been told the plan twists; the number that sizes the façade frame is 1.98, and on the symmetric plan reading the same 1.63 it would be 1.63. The stiffest path takes the load in a rigid-floor plan in proportion to stiffness times movement, and it is the movement relative to a torsion-free floor that decides it, not the movement relative to the average.

The near wall is told something wrong too. It moves 0.45 $u_0$ and carries less than half its direct share; the stiff core beside it is relieved by the twist. A ratio computed from the two edges averages a relieved wall with an overloaded one and reports their quotient, which describes neither.

The number the same analysis could give #

If the codes want a measure that needs only displacements an analysis prints, there is one that behaves. The floor’s displacement at the centre of rigidity is $u_0$ — that point, by definition, does not move when the floor rotates — so the worst edge’s displacement over the centre of rigidity’s is the amplification itself. The price is locating the centre of rigidity, which in a multi-storey building is a calculation rather than a reading. A cheaper proxy is a second run of the same analysis with no accidental eccentricity and the load at the centre of rigidity’s line, whose uniform translation is $u_0$; that is one more load case in a model that already has dozens.

The European tests avoid displacements altogether and ask for $e_n$ and $r$ directly, which is the same information from the other end: the amplification is $1 + e\,a/r^2$, a combination of exactly the two numbers EN 1998-1 tests. Either route gives a measure that grows with the thing it measures. The ratio of the edge drifts is the one route that does not.

The plan with the core in the middle #

The earlier essay’s worst plan had all its stiffness in a central core: no natural eccentricity, a torsional radius of 1.41 m on a 30 m floor, and an amplification of twelve from the accidental eccentricity alone. For that plan the ratio and the amplification agree, both 12.3, because a symmetric plan’s average does not move. So ASCE 7 flags it, correctly and loudly, and then the amplifier built from the ratio, $(12.3/1.2)^2 = 105$, is capped at 3. The plan the ratio reads most accurately is the plan for which the amplifier is least accurate, and the cap is doing all the work.

That is the other end of the same story. The ratio is honest for symmetric plans and dishonest for eccentric ones; the amplifier’s square exaggerates the honest readings and its cap then cuts them back; and between the two the plans that get their torsion amplified by close to the right amount are the moderately eccentric, moderately flexible ones in the middle of the map — which is where the calibration was presumably done.

The amplifier with a ceiling #

The ratio is not only a classification. ASCE 7 also builds an amplifier from it, applied to the accidental torsional moment in a building flagged as irregular:

$$A_x = \left(\frac{\delta_{max}}{1.2\,\delta_{avg}}\right)^2, \qquad 1 \le A_x \le 3$$

The square is there to reflect how torsional response grows in a flexible plan, and the cap at 3 to stop it growing without limit.

Because the ratio saturates, the amplifier does too, and for any plan stiff enough in torsion it never reaches its own cap. At a torsional radius of half the length the amplifier cannot exceed 1.62, however eccentric the plan; at 0.35 of the length, 2.17. At the most flexible radius drawn it reaches 3, holds, and then falls — the most eccentric plans in that family get less amplification than moderately eccentric ones, while their corners move furthest of all. The amplifier’s cap of three is reached only by the plans flexible enough in torsion that the ratio had room to grow.

The same arithmetic, by hand #

For the core-and-façade plan the numbers can be checked with nothing but the straight line. The centre of rigidity is 10.8 m from the west edge, so $e_n = 15 - 10.8 = 4.2$ m, and the accidental eccentricity adds $0.05 \times 30 = 1.5$ m: $e = 5.7$ m. The torsional radius is 10.56 m, so $e/r^2 = 0.0511$ per metre. The far edge is $30 - 10.8 = 19.2$ m from the centre of rigidity and moves $1 + 0.0511 \times 19.2 = 1.98$; the near edge is 10.8 m the other side and moves $1 - 0.0511 \times 10.8 = 0.45$. Their average is 1.21, the ratio $1.98/1.21 = 1.63$, and the amplifier $(1.63/1.2)^2 = 1.85$.

The ceiling needs one line. The ratio is a mediant of 1 and $a/e_n$ — the far edge’s distance from the centre of rigidity over the middle’s — weighted by how much the rotation matters, so it can never exceed $a/e_n$. For the core-and-façade plan that ceiling is $19.2/4.2 = 4.6$, far above its 1.63, and the plan is nowhere near saturation. But the ceiling falls as the stiffness moves off-centre: at an eccentricity of 0.30 of the length it is $(15 + 9)/9 = 2.7$, at 0.45 it is $(15 + 13.5)/13.5 = 2.1$. The ratio rises towards a ceiling that is coming down to meet it, and once it is close, it can only follow the ceiling down.

Why this measure was chosen anyway #

The ratio has one great virtue: it needs nothing but two displacements from the analysis the engineer has already run. Centres of rigidity are ill-defined in a multi-storey building, where each storey’s walls and frames interact through the floors above and below — a wall and a frame tied together at every floor share storey forces in proportions that change up the height — and torsional radii need a stiffness matrix. Two edge displacements are printed by any three-dimensional analysis. The floor is a beam lying down found the centre of rigidity to be a sharp idea only when the floor is rigid, and the codes chose a measure that survives when it is not.

And the ratio was chosen to trigger something — more analysis, the accidental-torsion amplifier, a prohibition in the highest seismic categories — rather than to rank. For that purpose a measure that crosses 1.2 early and stays above it does its job. The difficulty is only in the reuse: the amplifier treats the ratio as a measure of severity, which past its peak it is not.

Where the model stops #

One storey, rigid floor, elastic walls. A tall building’s storeys twist differently, and the ratio is computed storey by storey from interstorey drifts; the saturation argument holds at each storey but the storeys’ centres of rigidity need not line up. A floor that is not rigid in plan does not rotate as a body at all, and the edge displacements then include the floor’s own bending.

Static force at the centre of mass. The dynamic response of a torsionally flexible building can be larger than any static eccentricity suggests, because a building whose torsional and translational frequencies are close couples the two motions in its modes, and splits the mass that moves together between a sway and a twist — the reason the earthquake asks for a displacement and a torsional mode supplies it at the corners. The codes’ factor of 1.2 in the ratio’s thresholds is partly an allowance for that, and the saturation argument does not reach it; the spectrum is not a load is the reminder that every static eccentricity is a stand-in for a dynamic response.

Walls that stay elastic. When the walls near the centre of rigidity yield first — which in an eccentric plan they often do, because they carry the most — the centre of rigidity moves towards the far side and the eccentricity falls. The elastic ratio then overstates the twist at collapse.

What the pictures cannot show #

That the ratio is computed from drifts in an analysis, and drifts depend on stiffness assumptions — cracked or uncracked walls, the stiffness of the floor, how the core’s coupling beams are modelled — that move the centre of rigidity by metres. Two walls that agreed to be one is a reminder of how much a coupling assumption changes what a wall is. A ratio near 1.2 or 1.4 is a classification made on numbers that are not known to the second decimal place.

Still open: the building that twists in its own mode #

Everything here is a static push. A building whose torsional radius is close to its mass’s own radius of gyration has a twisting mode at nearly the frequency of its swaying one, and under dynamic load the two share energy: the corner’s response then depends on the ratio of those frequencies, not on either reading of a static push. Whether the ratio of edge drifts from a response-spectrum analysis — where each mode’s drifts are combined before the ratio is taken — keeps the static ratio’s saturation or loses it, and whether a building with coupled modes can read regular statically and irregular dynamically, is the question that turns this one into the one the codes’ dynamic procedures are really answering.