The coupling beam asked for shear

The shear that moves the moments found that in a redundant beam, unlike a determinate one, shear deformation changes the forces. A stubby propped cantilever carries less moment at its wall and more in its span. The carry-over factor every hand method passes along, a half, falls to nothing at a shear-flexibility parameter of 2 and then changes sign. That essay pointed at where the effect should be largest in a real building: the coupling beams of a pair of shear walls, the stubbiest members there are, whose shear flexibility should decide how much of the overturning moment the pair carries as a couple and how much each wall carries alone.

Two walls that agreed to be one had already built the pair. Twenty storeys of two 6 m walls either side of a 2.4 m opening, coupling beams 350 mm wide and 600 mm deep at every floor, carry 63% of their overturning as a push-pull couple between the walls rather than as bending in either, and the coupling beams are the hardest-worked members in the building. Its beams included their shear deformation. The question left was what that term is worth, and what a beam does when it is asked for more shear than it can give.

A beam two metres deep, a third as stiff #

A coupling beam is a short deep beam built into a wall at each end. Moved up at one end relative to the other, it resists in bending, with a stiffness $12EI/l^3$ for its clear span $l$, and in shear, with a stiffness $GA_s/l$. The two act in series, so the beam’s stiffness is

$$k = \left(\frac{l^3}{12EI} + \frac{l}{GA_s}\right)^{-1},$$

and the shear term matters in proportion to $(d/l)^2$. For the 600 mm beam over the 2.4 m opening, a span four times its depth, the bending stiffness is 164 MN/m and the shear stiffness 911; together, 139, 15% below bending alone. For a 2 m beam over the same opening the bending stiffness has grown with the cube of the depth to 6,076 MN/m while the shear stiffness has grown only in proportion, to 3,038, and the shear term dominates: the beam is a third as stiff as bending says.

And the walls barely notice. The beam’s stiffness falls to a third, and the pair’s degree of coupling — its base couple over its overturning moment — falls by under 3%. The largest beam shear falls by about 6%. The shear term cuts the beam’s stiffness, not its share of the work.

Why a stiff beam’s stiffness stops mattering #

The degree of coupling is set by a competition between the beams and the walls. The beams try to make the two walls bend as one; the walls resist through their own axial stiffness, since a couple between them means one wall stretching and the other shortening. While the beams are flexible, they are the weak link and their stiffness decides the coupling. Once they are stiff enough, the walls’ axial flexibility is the weak link, and making the beams stiffer, or softer within a wide range, moves the coupling little. The earlier essay found the same saturation from the other side: going from a 400 mm to a 1 m beam raised the coupling less than the cube of the depth suggested.

That is the paradox at the centre of the question. Shear deformation is large only in a deep beam, and a deep beam is the one whose stiffness no longer decides anything. In a slender beam, over a wide opening, the beam’s stiffness matters to the coupling, and there the shear term is a few per cent. The two never coincide. The redundant beams of the earlier essay moved their moments by tens of per cent because the stubby beam was the whole structure; in a coupled wall it is one link in a chain whose weakest link is elsewhere.

Up the height the picture is the same. The beams’ shears, highest about a third of the way up as the earlier essay found, move by a few per cent with the shear term at every level, for a 600 mm beam and for one twice as deep. A deeper beam shifts its peak a couple of storeys lower and carries a little more, because it couples the walls more tightly near the base. An analysis that leaves out the shear term gets the distribution right and the amounts within a few per cent.

Asked for little more, able to give much more #

So shear deformation is not what limits a coupling beam. What limits it is the shear it can carry.

With 60 kN at every floor, the largest beam is asked for 222 kN at a depth of 600 mm, 299 at 1.2 m and 328 at 2 m: the demand grows only as fast as the coupling does, and the coupling is saturating. The shear a beam can supply grows in proportion to its depth. At a limiting shear stress of 2 N/mm² — the order of what a conventionally reinforced deep beam can be relied on for — a 600 mm beam supplies 420 kN; at 5 N/mm², the order a beam with diagonal reinforcement reaches, 1,050. Making a coupling beam deeper buys capacity much faster than it buys demand.

The storey force drawn is a modest one, a total of 1,200 kN at the base of a twenty-storey pair. An earthquake asks for several times that, and the question is what the beams do when the multiple of the storey force is large enough to exceed what they can give.

The push #

The pair is pushed: the storey force at every floor raised step by step, and at each step every beam that has reached its capacity replaced by its capacity — a fixed pair of forces between the walls, of whatever size it has reached, with no further stiffness. The walls stay elastic, so the push isolates what the beams do.

The first beam to yield is near the third-height peak, at 1.9 times the storey force for beams limited to 2 N/mm². Yielding spreads up and down from there until, at 3.4 times, all twenty have reached their capacity. From that point the couple between the walls cannot grow: it is the sum of the beams’ capacities times the lever arm between the walls. Every further increment of overturning is carried by the two walls bending on their own, and the pair’s lateral stiffness falls to that of two cantilevers again — the slope of the push past the last yield is the two walls’ alone. Beams limited to 5 N/mm² hold out to 4.8 and 8.3 times the storey force before the same thing happens.

The coupling the beams’ capacity allows #

While the beams are elastic the coupling is constant at 0.627, whatever the force. Once they start yielding it falls, and once all have yielded it falls as one over the force, because the couple is fixed and the overturning is not. Beams limited to 2 N/mm² give a coupling of 0.44 at four times the storey force and 0.17 at ten; beams to 5 N/mm², 0.63 and 0.44.

So in an earthquake that pushes the pair well past its elastic range, the degree of coupling is a choice the designer makes with the beams’ capacity, not a property of their stiffness. The coupled wall’s plastic overturning resistance is the walls’ own plastic moments plus the beams’ summed capacities times the lever arm, and the elastic degree of coupling — which the shear term barely moves — is an elastic answer to an inelastic question. This is the reasoning behind designing a structure for its capacity rather than its elastic forces, applied to the one member type a coupled wall is designed around, and it is what a pushover of a coupled wall is for.

The rotation instead of the shear #

A beam that has given all its shear is not finished. It keeps the walls connected at its capacity and is moved further and further, one end up and the other down, as the walls rotate apart beneath it. The largest beam, limited to 2 N/mm², has a plastic chord rotation of 0.008 radians at four times the storey force and 0.039 at eight. That rotation is the real demand on a yielded coupling beam. Whether the beam can supply it is a matter of its reinforcement and not of its stiffness: tests of coupling beams with diagonal reinforcement sustain rotations several times larger than those of conventionally reinforced beams of the same size, which shear and slide at their ends long before.

The rotation is set by the walls’ sway, not by the beam. Every yielded beam at every level is moved by roughly the walls’ rotation times the lever arm over its span. A deeper beam of the same span sees the same rotation, so depth buys shear capacity but not rotation capacity. A wider opening, a longer beam, spreads the same relative movement over more length and asks each beam for less rotation. That is why openings that coupling beams span are sometimes widened rather than the beams deepened: the beam’s job past yield is to bend through a rotation, and length makes that easier where depth does not.

A wider opening, or a narrower one #

The beam’s span is the opening, and the opening is the architect’s before it is the engineer’s. With beams limited to 2 N/mm², the three choices of opening a corridor or a door line usually leaves behave in opposite ways in the elastic range and past yield.

A narrower opening, 1.5 m, makes the beams stiffer and the coupling higher, 0.68 against 0.63, and asks the largest beam for 293 kN at the base storey force instead of 222. It yields first, at 1.5 times the storey force, and all its beams have yielded by 3.0. Past yield its beams are asked for the most rotation: 0.059 radians at eight times the storey force, because the walls’ rotation apart is shared out over a span only 1.5 m long. A wider opening, 3.5 m, couples less, 0.56, asks its largest beam for only 159 kN, holds out to 2.7 and 5.2 times the storey force, and at eight times asks its beams for 0.027 radians.

The opening that couples the walls best is the one that asks its beams for the most, in both senses: more shear while they are elastic and more rotation once they are not. The top sway hardly differs between the three at large multiples, 590 to 650 mm at eight times the storey force, because past yield every pair is two cantilevers held together by a capped couple. The choice of opening is a choice between a stiffer pair in a wind and a more forgiving one in an earthquake. A deep beam over a short opening is the member that ends up neither bending nor shearing as a beam does but carrying its shear as a strut between two corners, which is why it is detailed with the diagonal bars a strut-and-tie model would give it.

The pair, by hand #

The 600 mm beam over 2.4 m: $I = 0.35 \times 0.6^3/12 = 0.0063\ \text{m}^4$, so $12EI/l^3 = 12 \times 30 \times 10^6 \times 0.0063/13.8 = 164\,000$ kN/m. The shear stiffness, with $G = 12.5$ GPa and $A_s = \frac56 \times 0.35 \times 0.6 = 0.175\ \text{m}^2$, is $GA_s/l = 911\,000$ kN/m, and the series stiffness 139,000.

The pair’s base overturning under 60 kN at each of twenty floors 3.2 m apart is $60 \times 3.2 \times 210 = 40\,320$ kN·m. With a degree of coupling of 0.627, the couple is 25,260 kN·m, a base axial force of $25\,260/8.4 = 3\,010$ kN in each wall, which is the sum of all twenty beams’ shears: 150 kN each on average, 222 at the peak.

At 2 N/mm² each beam can give $2 \times 350 \times 600 = 420$ kN, so twenty beams give 8,400 kN at most, a couple of $8\,400 \times 8.4 = 70\,600$ kN·m. At ten times the storey force the overturning is 403,000 kN·m, of which the couple is 17.5%: the 0.17 the push gives.

Elastic walls, a capped beam, one direction #

The walls stay elastic. Real walls crack and their bases yield, which softens the walls’ own stiffness and moves the coupling back up for a while, until the walls’ plastic moments are reached as well. The push here isolates the beams.

Each beam is elastic and then perfectly plastic in shear. Real coupling beams lose stiffness as they crack, before yield, which spreads the yielding more evenly up the height. Diagonally reinforced beams harden; conventionally reinforced deep beams can lose strength abruptly at their ends.

One direction of push. An earthquake reverses the push every half cycle and the beams’ rotation demands are cyclic. Deep conventionally reinforced beams slide at their ends under reversal, which is the failure diagonal reinforcement exists to prevent.

The beams’ shear capacity is a single stress. A limiting shear stress over the section is how codes bound a coupling beam’s strength; it stands for a reinforcement layout that would have to be designed to deliver it.

What the shear term is for #

The deflection that is not bending introduced the shear term as a correction to the deflection of short deep beams, and the stiffness that belongs to the span showed that it makes a member’s stiffness a property of its span rather than its section. The redundant beam showed it moving forces. The coupled wall shows its limit. It moves forces only where the stubby member is the structure’s weak link, and a coupling beam deep enough for the term to matter is never the weak link while it is elastic. Its weakness is its shear capacity, and once that is reached its stiffness of any kind is irrelevant.

The same reasoning disposes of a larger correction than the shear term. Design practice reduces a coupling beam’s stiffness for cracking to a fraction of its gross value, often a fifth or less, which is a far larger cut than shear deformation makes to an uncracked one. It moves the elastic coupling more, because a beam a fifth as stiff can fall back into the range where its stiffness matters. But it moves the plastic coupling not at all, and it changes when the beams yield rather than whether they do. What a cracked-section stiffness buys is a better estimate of the drift at which the beams start yielding, which is the quantity a design for rotation needs.

That makes the coupling beam an instance of a general rule about how a tall building stands still: the elastic share of the load between a building’s lateral elements is set by their stiffnesses, and their inelastic share by their strengths, and the second is the one an earthquake asks about. A designer chooses the coupling a pair of walls will have in an earthquake when choosing how much shear its beams can give, and the displacement the earthquake asks for is then delivered as rotation in those beams.

Still open: the wall that yields at its base while the beams are yielding #

The walls here never yield. A real coupled wall is designed for its beams to yield first and its walls’ bases to yield last, so that a mechanism forms with every beam rotating and both walls hinged at the ground. Between the two the walls’ bases crack and soften, and a softer wall rotates more for the same overturning, which asks each beam for more rotation before the walls have reached their own strength. Whether a pair whose walls soften as they are loaded asks its beams for a rotation the beam’s capacity was not sized for — and whether the beams that yield first, near a third of the height, are asked for most of it while the ones near the top are still elastic — is the question of how a coupled wall’s mechanism forms rather than whether it does.