Structural Engineering & Statics
where the load goes
The collection

Every essay — page 17

Essays 385 to 396 of 396, in the same order.
The worst section of a haunched rafter is inside the haunch. Utilisation along a 15.3 m portal rafter carrying 8 kN/m, with an eaves moment of 500 kNm and an apex moment of 150, haunched over 3 m from 906 mm deep down to the rafter's own 453. The moment is largest at the eaves and the depth is largest there too, so the eaves is at 0.42; the apex is at 0.31. The peak is 0.46 at 2.98 m — the haunch tip, where the section has just become the bare rafter and the moment is still 226 kNm. The dashed curve is the same rafter with no haunch, which reaches 1.02 and does not pass. Sections and stress

The section that governs is inside the haunch

A tapered cantilever has its worst section somewhere along it because the moment grows linearly and the modulus quadratically. A haunched rafter has the same competition with a step in it, and the step is where the check lands — at neither end of the member, at a station no formula names.

6 figures · Tapered member What a damper at the anchorage can do, and the ceiling it cannot pass. Modal damping against damper size for a 200 m stay at 4500 kN, with the damper 4 m from the anchorage — 2.0 per cent of the length. Each curve is a mode, found as a complex root of the taut string with a viscous damper in it rather than from a formula. Every one of them peaks at 1.00 per cent of critical, which is x/2L exactly, and the peaks are at different damper sizes — a higher mode wants a smaller damper, because it moves faster at the same amplitude. The curves are flat near their peaks: half the optimum coefficient gives 80 per cent of the ceiling, and so does twice it. Dynamics

The damper that is too near the end

A stay cable has almost no damping of its own, so it is given a damper — and the damper cannot go where the motion is, because the middle of a two-hundred-metre stay is a hundred metres above the road. What it can supply is then decided by one length, and no amount of damper changes it.

5 figures · Cable dynamics Damping that is computed rather than measured. Radiation damping of a 2.5 m block on soil with a shear wave speed of 178 m/s, against how heavy the block is made — the horizontal axis is a multiple of the 150 tonne block drawn. At that mass the three modes are at 47, 29 and 13 per cent of critical. None of this is dissipation: the energy leaves as waves travelling away into the half-space, so the quantity is a geometrical coupling and it can be computed from the size of the footing, the density of the soil and its shear modulus. Every curve falls as the block gets heavier, because a heavier block moves less for the same wave field — which is the one counter-intuitive thing here: mass buys frequency and costs damping. Dynamics

The damping that is radiated

Every response in this field is quoted with a damping assumption attached, because damping is measured rather than designed and the measurement has a factor of two in it. A machine block on the ground is the exception: its damping is not dissipation at all, and it can be computed from three numbers none of which is a material property of anything that dissipates.

5 figures · Vibration isolation 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. Dynamics

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.

5 figures · Pounding The same bridge by the two theories it might have been designed by. Deck moment along a 500 m suspended span with half of it loaded, computed twice. Elastic theory treats the deck as a beam and applies the cable's extra tension as an upward load: 80.1 MN·m. Deflection theory keeps the cable's total tension acting on the deck's own deflected shape — a geometric stiffness — and returns 56.3, which is 30 per cent less. The extra cable tension is very nearly the same in both (6.20 against 6.11 MN), so nothing about the cable is in the difference: it is entirely the H·v″ term the older theory drops. Structural form

The tension that was left out

A suspension bridge's deck sits on a cable pulling hard along it, and a member with a large tension in it is stiffened by that tension. Leaving the term out of the deck's own equilibrium is what elastic theory does, and on a long span it asks for fourteen times the girder.

5 figures · Stiffening girder What the second, third and fourth arms are worth. Top drift removed against the number of outriggers, each arrangement at its own optimum levels, on a 40-storey core 200 m tall. One arm at 59 per cent of the height removes 82.3 per cent of the drift. A second, with both moved to 35 and 71, takes it to 91.6 — a gain of 9.3 points, which is half of what was left. The third is worth 3.0 and the fourth 1.5, and each one costs a storey of the building's most valuable height. Structural form

What the second arm is worth

One outrigger at its best height removes five sixths of a tall core's drift, which sounds like the end of the argument. A second removes half of what is left, a third half of that, and each of them costs a storey of the most valuable floor area in the building — so the question is not where to put an outrigger but how many the arithmetic still justifies.

6 figures · Outrigger Two restraints, and only one of them cares how much was imposed. Crack width against the restrained strain, for the same 20 m wall restrained two ways. Edge restraint — a wall cast on a base — gives a width proportional to the strain, because the concrete has to accommodate the movement and the cracks are where it does. End restraint — a bay cast between two that have hardened — gives 0.32 mm at every strain on the axis, because the crack opens only until the steel can push the cracking force back into the concrete, and that force is a property of the section. At the 304 microstrain this wall is asked for, the two are 0.17 and 0.32 mm, and the movement is divided into 35 cracks and 19. Internal forces

The bay that is cast last

A wall cast on a base cracks in a way the steel controls: more movement gives wider cracks, and reinforcement decides how many. A bay cast between two walls that have already hardened cracks in a way that has nothing to do with how much movement there was — and the threshold that separates a controlled crack from a single wide one is a quantity of steel rather than a limit on anything.

5 figures · Restraint cracking Two load sets with the same resultant and different work. The two ways of putting a uniform load of 10 kN/m onto a beam element 6.00 m long. Both put 30.0 kN at each node, so both have the same resultant and the same moment about any point — they are equivalent for a rigid body. The consistent set adds a couple of 30.0 kN·m at each end, in opposite senses, which is what makes it do the same virtual work over the element's shape functions as the real load does. The couples cancel in the resultant, which is exactly why the resultant cannot see them, and they are the whole difference between an exact answer and one that is a third out. Equilibrium

Equivalent in work, not in resultant

Two force systems with the same resultant and the same moment about every point are interchangeable — for a rigid body. A finite element is not a rigid body, and substituting one for the other on a beam element leaves the tip of a cantilever a third too low with no warning of any kind.

5 figures · Force couple The part of a base plate that is delivering anything. A plan of the plate: the 260 × 260 column in the middle, and the shaded region it can reach — its own outline grown by 43 mm, which is t√(f_y/3f_jd) for a 20 mm plate on grout at 20 N/mm². That is 78176 mm² of a 200000 mm² plate, or 39 per cent of it. The corners are outside it and are carrying nothing: a plate cantilevers from the column's perimeter and runs out of bending capacity at c, so making it larger in plan beyond that changes nothing at all, and making it thicker changes everything. Connections

The plate that is only as big as it is thick

A base plate's design is presented as a bearing calculation: an area, a bearing strength, and a check that the pressure fits. The area in that calculation is the plate, and the plate cannot deliver it — a 20 mm plate on a 500 square reaches 43 mm past the column and the corners carry nothing at all.

5 figures · Base plate Force and capacity round a weld group, which do not vary together. Utilisation round the c shape group under 100 kN at 150 mm, walked from one end of the weld to the other, with the force scaled onto the same axis for comparison. The capacity is 1460 N/mm where the force runs along the weld and 1789 where it runs across it — √1.5 more — so the utilisation is not simply the force in different units. The worst point is at 47 degrees to the weld, where the capacity is 1612 N/mm, and the group carries 182 kN against the 165 the along-the-weld value would allow. Connections

The weld that is stronger where it is pulled

A fillet weld pulled across its axis is √1.5 stronger than the same weld pulled along it. On a group under an eccentric load the direction of the pull varies from point to point, so the capacity does too — and the gain that follows is worth twenty per cent at one eccentricity and nothing at all at another.

5 figures · Weld strength Where the span moment and the support moment cross. Span moment and support moment against the joint's stiffness, for a 6 m beam under 30 kN/m. They move in opposite directions because they add to a constant — the simple-span 135 kN·m is fixed by statics and the joint only decides how it is split. They cross at 68 kN·m, where the joint is delivering 75 per cent of the fixed-end moment, and neither the crossing nor the fraction depends on the beam, the span or the load: it is the point where f·wL²/12 equals wL²/8 − f·wL²/12, which is f = 0.75 for every beam there has ever been. Connections

The joint that was chosen

A joint's stiffness decides how a beam's moment divides between its span and its supports, and the two add to a constant. So there is a stiffness at which they are equal, the beam is sized by the smaller of two numbers rather than the larger of one, and the design moment is half what a simple connection leaves behind.

5 figures · Joint classification The deflection a building can take, and which way it is bending. The limiting deflection ratio — the sag or hog of a building divided by its length — at a critical tensile strain of 0.075 per cent, against how long the building is compared with its height. Two curves, and the gap between them is the whole finding: hogging is worse than sagging by exactly 2.0 times, because the neutral axis of a hogging building sits near the bottom and the tension face is the full height of the wall. The minimum is at L/H = 1.6, where the two mechanisms cross: a squat building cracks diagonally in shear and a long low one cracks in bending at the extreme fibre. Deflection

The crack is a strain, not a slope

Angular distortion is what every table of settlement limits is written in, and it is a proxy. What cracks a building is a tensile strain in it, and reading the building as a deep beam says which of two mechanisms supplies it, why hogging is exactly twice as damaging as sagging, and why a horizontal stretch of a few millimetres takes half the capacity away before the settlement starts.

5 figures · Differential settlement