Held by something that goes soft

A guyed mast is the cheapest tall structure there is. A slender lattice column two metres across can carry an aerial 300 metres up, on a fraction of the steel a self-supporting tower would need, because it is not resisting the wind by bending — it is being held sideways at intervals by wires.

The wires are the problem. A wire has no bending stiffness at all, and what it offers the mast is not material stiffness but geometric: it resists movement by having to change shape, and how much it resists depends entirely on how tight it already is.

Which free body produced the number #

Cut a guy and take the mast. What the guy applies to it is a force along the chord, of magnitude $T$, whose horizontal component is $T\cos\theta$ and whose vertical component is $T\sin\theta$ — pointing down, into the mast, because the guy is anchored to the ground.

The mast moves sideways by $u$ at that level. The chord’s extension is $u\cos\theta$ — the movement projected onto the guy — and the tension that goes with it satisfies

$$T = T_0 \pm \frac{E_{\text{eff}}(T)\,A}{L_c}\,u\cos\theta$$

which is nonlinear, because $E_{\text{eff}}$ is the tangent modulus at the tension being solved for. Solving it by stepping is what makes a slackening guy oscillate: at a small tension the cable is so soft that a step returns the pretension in full, and the next step takes it away again. Solved by bisection instead, the leeward equation always has a root strictly above zero — which is the physical statement that a cable does not go slack in one movement. It softens as it loses tension, so the tension it can lose shrinks with it, and what actually happens is a guy hanging in a deeper and deeper curve at a tension approaching nothing.

The horizontal restraint the pair supplies is then $(T_w - T_l)\cos\theta$, and the check on all of it is statics: the base is a pin, so it can take no moment, and the guys’ moments about it have to add to the wind’s. On the mast drawn they do, to a part in fifty.

Ernst’s correction, and why it is a cliff #

The tangent modulus of a sagging cable is

$$E_{\text{eff}} = \frac{E}{1 + \dfrac{w^2 L_h^2 A E}{12\,T^3}}$$

and the $T^3$ is the whole of it. At 20 per cent of breaking load a guy has 99 per cent of its material stiffness; at 10 per cent it has 90; at 5 per cent it has 53; at 2 per cent it has 6.

That is why guys are pretensioned at all. Not to hold the mast — a mast with no wind on it needs no holding — but to put every guy on the flat part of that curve before the wind arrives, so that the leeward one is still a spring when it is being asked to be one.

The same argument, one level up, is the cable whose stiffness comes from its shape. What a guyed mast adds is a sign: the guy the mast is moving towards is the one losing tension, so the support on the side the structure is heading for is the one that gives way first.

Doubling the wind more than doubles the sway #

A structure whose supports get softer as they are loaded is nonlinear in a direction nobody wants. On the mast drawn, a wind of 1.0 N/mm moves the top 32 mm; 3.0 moves it 100; and 6.0 moves it 226 — so doubling the design wind multiplies the sway by 2.26.

That is a mild nonlinearity by the standards of this field and a severe one by the standards of a serviceability check written as a linear extrapolation. It also means the return period matters more than usual: the 50-year wind does not move the mast fifty-year-linearly more than the annual one.

Two effects compound to produce it. The leeward guys are softening as the wind rises, and the windward guys’ tension is rising, which pulls the mast down harder — so the axial load, and with it the second-order amplification, grows with the wind that is causing the deflection.

The axial load the mast carries is the vertical component of every guy above the section, so it accumulates downward and the mast becomes a column that has been loaded by its own restraints — which is a load that makes itself worse at the same time as it is the thing holding the mast up.

The mast is a continuous beam on springs, and it bulges #

With three guy levels the mast is a beam-column on three lateral springs, and its deflected shape is not the smooth curve a cantilever gives. It is a continuous beam’s shape: hogging at each guy level, sagging between, with the largest movement not necessarily at the top.

On the mast drawn, the mid-height guy level moves 123 mm and the top moves 100 — the mast bulges between guys rather than leaning. The worst bending moment is 554 kNm and it is at the lowest guy level, not at the base and not at the top.

That is the structural reason guy levels are spaced the way they are. Each span between guys is a beam-column, its buckling load is $\pi^2EI/\ell^2$ against the longest of them, and the mast carries 801 kN of axial load put there entirely by the guys. Spacing the guys further apart weakens each span quadratically; spacing them closer adds anchors, and an anchor is a foundation in a field a hundred metres from the mast.

A mast is also a beam on discrete springs rather than a beam on supports, and a support that is a spring rather than a point changes the shape of the whole moment diagram above and below it.

The pretension window, and both of its edges #

Guy pretension is the one number a designer of a mast actually chooses, and both directions are punished.

Too little and the guys are on the steep part of Ernst’s curve: at 2 per cent of breaking load the top moves 195 mm instead of 88, because at design wind the leeward guys have a tangent modulus of eight per cent of the steel’s and are supplying under a tenth of the pair’s stiffness.

Too much and the guys’ vertical components add up. Every kilonewton of pretension puts $\sin\theta$ kilonewtons into the mast — and $\sin\theta$ is large, because a guy at a 60 m radius on a 120 m mast leaves the top at 63 degrees to the horizontal. At 30 per cent of breaking load the three levels together are pushing 2,160 kN down a member whose Euler load between guy levels is 2,467.

The received range — eight to fifteen per cent of breaking load — is that elbow, and it is not a rule of thumb so much as the place where two curves of very different shape cross a designer’s patience.

Three guys, and the wind between two of them #

The section drawn in the first figure is a two-dimensional slice through something that is not two-dimensional. A real mast has three guys at each level, at 120 degrees in plan, because three is the smallest number that restrains a point in a plane and because a mast that is a triangle in cross-section wants its guys on its corners.

That changes the worst case. With the wind along one guy, the two others share the restraint between them and each contributes $\cos 60° = \tfrac12$ of its horizontal pull in the wind’s direction — so the pair supplies the same total as one guy would, and the guy directly upwind is being unloaded on its own. With the wind between two guys, one guy takes the whole leeward duty and two share the windward one.

The two arrangements differ by about fifteen per cent in the restraint they offer, and by rather more in the tension the most loaded guy sees. Since the wind comes from anywhere, the mast is designed for the worse one — which means the arrangement that looks symmetric in plan is not symmetric in its response, and there is no orientation that avoids it.

And the vertical components do not care about direction at all. Three guys at 160 kN each put $3 \times 160 \sin\theta$ into the mast whatever the wind is doing, which is why the axial-load side of the pretension trade is worse in three dimensions than the drawing suggests.

Each level is itself three tension members meeting at a point in three dimensions, which is the arrangement six equations describe and a drawing shows three of.

The guy is also a mass on a string #

Everything above is static, and a mast is not. The guys have mass, they sag, and a sagging cable has natural frequencies of its own — low ones, because the restoring stiffness is geometric and the span is long.

Two consequences follow. The guys can resonate with the mast, coupling a global sway mode to a cable mode at a frequency neither would have alone; and the guys can resonate with the wind, since a cable in cross-flow sheds vortices at a frequency proportional to the wind speed divided by the cable diameter, which for a 40 mm strand at ordinary wind speeds is squarely in the range of its own modes.

The second is why guys have damping devices and helical strakes on them, and why the failures that have happened to guyed masts have more often been about a guy than about a mast.

A guy is also exposed to a mechanism of its own: a slender cylinder in a flow sheds vortices at a frequency it chooses rather than one the wind supplies, and if that frequency finds the guy’s own it is wind that brings its own frequency.

The mast carries its own weight into the same check #

One term has been left out of the axial load and it is not small. A lattice mast 120 metres tall weighs something of the order of a hundred tonnes, and all of it arrives at the base through the same member that is carrying the guys’ vertical components.

That matters because the two accumulate in the same direction. At the base the mast is carrying its own weight plus every guy level’s vertical pull, so the section that is worst in bending — the lowest guy level, where the moment peaks — is not the section that is worst in compression, which is the foot. A guyed mast is therefore checked at two stations for two reasons, and the proportions of the two checks move in opposite directions as the pretension changes: more pretension is more axial load everywhere and less bending at the guy levels.

It also explains a detail of erection that looks like superstition. A mast is built and guyed in stages from the bottom up, and each stage’s guys are tensioned before the next is lifted — so the lower part of the mast spends the construction period carrying pretension it was designed for and no wind, which is the one load case in its life where the two are not in proportion.

Ice moves both curves the wrong way #

The load case that actually governs most guyed masts is not on any figure here, and it acts on the one term Ernst’s correction is most sensitive to.

Twenty-five millimetres of radial ice on a 40 mm strand makes it 90 mm across. The ice itself is

$$\frac{\pi}{4}\left(0.09^2 - 0.04^2\right) \times 900 = 4.6\ \text{kg/m}$$

against the strand’s own 9.4, so the guy’s weight goes from 90 N/m to 135 — a factor of 1.5, which enters Ernst squared:

$$\frac{E_{\text{eff}}}{E} \;=\; \frac{1}{1 + 2.25\,\alpha}$$

where $\alpha$ was the bare guy’s sag term. At 10 per cent of breaking load that takes the tangent modulus from 90 per cent to 80; at 5 per cent it takes it from 53 to 33. The cliff has moved to the right, and the pretension that put every guy on its flat part no longer does.

The same 2.25 appears again in the other direction. Drag goes with the projected area, and 90 mm of iced cable presents 2.25 times the area of 40 mm of bare one — so the guys’ own wind load, which is a substantial fraction of a mast’s total, more than doubles at the same wind speed. The structure is being asked for more restraint by softer springs.

And there is a third term that is not a number at all. Ice does not shed evenly: one guy clears in the sun and the other two do not, so a level that was assumed symmetric acquires an out-of-balance pull of a few kilonewtons, applied suddenly, to a structure with about half a per cent of damping. The shedding is a dynamic load case with an impulse in it, and it is the mechanism behind more guyed-mast damage than wind alone.

Which is why the design ice case is not simply “the wind case with heavier guys”. It is a different combination — usually a lower wind with ice, checked separately — and on a mast in a cold climate it is the case that sizes the guys.

Losing one guy is losing the mast #

The mast has been treated as a beam-column on three springs. It is worth asking what it is on two, because the answer is not a reduction in capacity but the absence of one.

Take out a whole guy level and the buckling length between the remaining ones doubles. The Euler load carries $\ell^2$ underneath, so

$$P_{cr} = \frac{2{,}467}{4} = 617\ \text{kN}$$

against an axial load of 801 kN — put there, as the essay has said throughout, by the guys themselves. The mast does not become slender; it becomes a column past its critical load, and it goes at once.

Losing one guy of the three at a level is very nearly as bad, and the reason is that cables pull and do not push. Two guys at 120° can hold the mast against a force pushing it away from both of them, by taking more tension. They cannot hold it against a force pushing it toward the missing anchor, because doing so would require the remaining pair to push. So one broken guy leaves a 120-degree sector of the compass in which that level is not restrained at all — and the wind comes from anywhere.

That is a structure with no alternative load path whatever, and it is worth being blunt about how far it is from the rest of this collection. A frame that survives losing a member does so because there is a second route; a guyed mast has one route per direction and no second one anywhere. Its robustness is not designed in — it is bought entirely by making the guys unlikely to fail: corrosion protection, a generous factor on the strand, anti-vandal sleeves at the bottom, and inspection.

It also puts the pretension window in a different light. The lower edge of that window was justified above by stiffness. The upper edge is now doing two jobs: it limits the axial load in a member whose failure is a collapse, and every kilonewton of it is a kilonewton closer to the 617 that one lost guy level would leave behind.

Where the model stops #

The wind is a static pressure. It is not: it is a spectrum, with a mean part the analysis above is honest about and a fluctuating part that a mast — lightly damped, low frequency, and nonlinear — responds to in a way that a gust factor represents rather than computes.

The guys are two per level. Real masts have three at 120 degrees, so the windward and leeward pair drawn here is a section through an arrangement that also has to resist wind from any direction, and the worst direction is between two guys rather than along one.

Ernst’s correction assumes a parabolic sag and a taut cable. Both are safe at working tensions and neither is at very low ones, which is exactly the region the essay is about. The curve at two per cent of breaking load is qualitatively right and quantitatively an extrapolation.

And the anchors do not move. A guy anchor is a mass of concrete or a rock bolt resisting an inclined pull of several hundred kilonewtons; a few millimetres of movement there is a few millimetres of extension in the guy, which at these stiffnesses is a real loss of pretension.

What the pictures cannot show #

The deflection in the first figure is drawn at more than five hundred times its real size. A mast moving 100 mm over 120 metres is out of plumb by less than a tenth of a degree, and would look perfectly straight from the field the anchors are in.

Nor can any figure show the operation that sets the number this whole essay is about. Guy pretension is put in by tensioning each guy in turn, at a temperature, against a load cell or a measured sag — and then the temperature changes. A steel guy 134 metres long loses about 40 kN of tension for a 20 °C rise, which on a 160 kN pretension is a quarter of it, and it happens every summer afternoon.

The assumption the figure rests on #

Every number here assumes the mast is plumb and the three guys at a level are equally tensioned.

Neither survives contact with a site. A mast erected out of plumb has a permanent lateral force from the guys’ own pretension, which its own analysis has no load case for. A level with unequal tensions has a resultant that pushes the mast sideways before any wind arrives. Both are corrected by adjustment, and adjustment changes with temperature, with creep in the strand, with settlement at an anchor, and with the last person to climb the mast.

So the initial state of a guyed mast is not a design decision, it is a maintained condition — and that is the real difference between this structure and every other one in this collection. The others are built and then stand. This one is built, tuned, and re-tuned, and its analysis is only as good as the last measurement of the tensions it assumes.

The real limit of a prestressed system is not a stress anywhere. It is a family of members going slack — the stiffness that comes from the shape is only there while every member in the family is still pulling.