Structural Engineering & Statics
where the load goes
Structural form

Held by something that goes soft

A guy is a cable, so it has no stiffness of its own — what resists a mast's movement is the guy's geometry changing, and how much of that there is depends on the tension already in it. Wind pushes the mast towards the leeward guy, which is the one losing tension.

Assumes The stiffness that comes from the shape, Strong enough and still falls over and The load that makes itself worse.

19 min read 10 figures Geometry beats materialThe load must go somewhereThe statics of things that do not move

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.

The windward guy tightens, the leeward one gives wayA 120 m mast on three guy levels, at a wind of 3 N/mm, with the deflection drawn 0.5 times its true size. The guys start at 160 kN each and end at 259 against 95, 299 against 83, 221 against 112 kN. The leeward guys still carry a real force — the lowest keeps 28 per cent of its partner's tension — and supply almost none of the restraint, because their tangent modulus has fallen to 54 per cent of the steel's. The mast top moves 100 mm, its worst bending moment is 554 kNm at 40 m, and it is carrying 801 kN of axial load that nothing but the guys put there.259 kN95299 kN83221 kN112wind 3 N/mmtop moves 100 mmleewardwindwardaxial load from the guys 801 kN · deflection exaggerated 0.5×
Fig. 1 A 120 m mast on three guy levels at design wind. Every guy started at 160 kN. The windward ones are now at 259, 299 and 221 and the leeward ones at 95, 83 and 112 — still carrying a real force, and supplying between a third and a half of the pair’s stiffness, because a slack cable is soft long before it is loose.

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 TT, whose horizontal component is Tcos⁡θT\cos\theta and whose vertical component is Tsin⁡θT\sin\theta — pointing down, into the mast, because the guy is anchored to the ground.

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

T=T0±Eeff(T) ALc ucos⁡θT = T_0 \pm \frac{E_{\text{eff}}(T)\,A}{L_c}\,u\cos\theta

which is nonlinear, because EeffE_{\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 (Tw−Tl)cos⁡θ(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

Eeff=E1+w2Lh2AE12 T3E_{\text{eff}} = \frac{E}{1 + \dfrac{w^2 L_h^2 A E}{12\,T^3}}

and the T3T^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.

A slack guy is not a weak spring, it is barely a springErnst's tangent modulus — the stiffness a sagging cable actually offers, against the steel's own — plotted against tension as a fraction of breaking load, for a 60 m guy of 1200 mm². The correction goes as the cube of the tension, so the curve collapses rather than sloping: at 10 per cent of breaking load the guy has 88 per cent of its material stiffness, and at 3 per cent it has 20. That is why a leeward guy stops contributing long before it stops carrying load, and it is the whole reason a guyed mast is pretensioned at all.01020304000.20.40.60.81tension (% of breaking load)tangent modulus ÷ E88% at 10%20% at 3%the correction falls as the cube of the tension, so it is a cliff rather than a slope
Fig. 2 Ernst’s correction against tension, for the guys drawn. It is not a slope, it is a cliff: everything above about fifteen per cent of breaking load behaves as steel, and everything below about five behaves as string. The design range for guy pretension is the shoulder of this curve, and it is narrow because the curve is.

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.

Prestress buys a stiffness no change of material canFour cables of identical steel — 134 m, 1200 mm², E = 160000 MPa — differing only in the tension put into them before the load arrived. The initial stiffness is 8T₀/L exactly: 0.0, 2.4, 9.6, 38.2 kN/m at T₀ = 0, 40, 160, 640 kN, and no property of the steel appears in that expression. The slack cable leaves the origin flat — it has no stiffness whatever at zero load, and its sag grows as the cube root of the load, reaching 1.921 m under the same 12 kN that puts 0.314 m into the tightest of them. Four curves of one cable: the tightest starts 16 times stiffer than the slackest that has any stiffness at all, and every other property they share.00.511.52024681012midspan sag (m)total load on the cable (kN)T₀ = 0 kN · k₀ = 0.0T₀ = 40 kN · k₀ = 2.4T₀ = 160 kN · k₀ = 9.6T₀ = 640 kN · k₀ = 38.2all at 0.09 kN/mat zero prestress the curveleaves the origin flat
Fig. 3 The same geometric stiffness, in the structure it was derived for. Prestress buys stiffness that no change of material could, and the reason is visible in the expression: what is being made stiffer is a geometry, and tension is what a geometry costs.

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 load that makes itself worseThe amplification of a deflection against the ratio of applied load to buckling load. A structure at half its buckling load deflects twice as far as first-order analysis predicts, and the curve runs away well before the load is reached.00.20.40.60.80246810applied load ÷ buckling load1.3×1.5×2.0×3.3×first-order analysis says the answer is always 1×one over one minus the ratio
Fig. 4 The amplifier the axial load turns on. The mast here sits at 0.32 of its Euler load between guys at design wind, which multiplies the first-order deflection by about a half — and the axial load producing it is entirely the guys’ doing.

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 π2EI/ℓ2\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.

How much of a deflection belongs to the beamThe share of the total deflection that is the beam's own bending, against the stiffness of what it sits on. A 14 m beam on four supports under a uniform load: on rigid supports every millimetre is the beam's, and the share falls away as the supports soften until almost none of it is. The beam drawn beside this figure sits at 0% — so 100% of what it does is happening somewhere a beam calculation never looks. The two flexibilities are in series, which means the softer one governs and stiffening the other buys nothing.4020080030002000000.20.40.60.81stiffness of each supportfraction of the deflection that is bendingrigid supportsare over here0%the beam drawn
Fig. 5 The beam on discrete springs the mast is. A support that is a spring rather than a point changes the shape of the whole member, and a spring whose stiffness depends on the load it is carrying changes it again at every load level.

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⁡θ\sin\theta kilonewtons into the mast — and sin⁡θ\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.

Too little pretension and it sways; too much and the mast carries itTop deflection and mast axial load against guy pretension, at a wind of 3 N/mm. The deflection curve falls steeply and then flattens — below about ten per cent of breaking load the guys are too soft to be supports, and above it more tension buys almost nothing. The axial load rises without limit, because every newton of pretension has a vertical component and the mast is the only thing under it: at 30 per cent the guys are pushing 2160 kN down a member whose Euler load between guys is 2467 kN. The design window is where the first curve has flattened and the second has not yet mattered, and it is narrow because both are steep near it.051015202530050100150200pretension (% of breaking load)top deflection (mm)deflectionaxial load, to 2160 kNas built: 100 mmleast sway at 22% · Euler load between guys 2467 kN
Fig. 6 The two halves of the choice on one axis. The deflection curve has an elbow at about eleven per cent and is flat after it; the axial-load curve is a straight line rising without limit. The design window is where the first has flattened and the second has not yet mattered, and the reason it is a window at all is that the two curves have completely different shapes.

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°=12\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 3×160sin⁡θ3 \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.

Three legs, three equations, one answerA rigid top on three legs carrying 100 kN at (0.32, 0.18) m. The three equilibrium equations available — one vertical and two moments — leave three unknowns, so the system is exactly determinate and the reactions are 56.9, 9.2, 33.8 kN. Move the load anywhere and the answer moves with it; nothing about the legs' stiffness enters.56.9 kN9.2 kN33.8 kN100 kNthree legs · rank 3 · determinateΣV, ΣMx and ΣMy all close to 1e-14 kN
Fig. 7 Three members meeting at a point, in three dimensions. A guy level is exactly this: three tension members and a mast, whose equilibrium is a three-dimensional statement that a section through it can only ever half-answer.

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.

The cross-wind amplitude through the lock-in bandA 1.2 m cylinder at 1.2 Hz. Vortices leave it at St·U/D, so the critical speed at which they match the structure is 8 m/s — a breeze, met many times a year rather than once in fifty. At a Scruton number of 9.1 the peak amplitude is 222.7 mm, which is 19% of the diameter. 02468101214161820050100150200250wind speed (m/s)cross-wind amplitude (mm)0.50% damping — Sc = 9.1, peak 222.7 mm
Fig. 8 The mechanism a guy is exposed to. A slender cylinder in a flow sheds at a frequency it chooses, and a lightly damped structure near that frequency locks on to it — which is a poor description of a mast and an excellent one of the wire holding it up.

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.

The column curveFailure load against slenderness, as a fraction of the squash load. A stocky column crushes; a slender one buckles at the Euler load; the crossover is where the two curves meet, and real columns fall below both near it.2040608010012014016018000.20.40.60.811.2slenderness (effective length ÷ radius of gyration)they cross at λ = 76squashingEuler bucklingreal columns, which are neither
Fig. 9 The check the accumulated axial load is made against. A mast between guy levels is an ordinary column of that length, and everything above the level in question — its own weight and every guy’s vertical pull — is what it is carrying.

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 net is nearly linear right up to the moment half of it lets goLoad against centre deflection for a 30 m square net of cables at 2 m centres, a sagging family 2.4 m deep and a hogging family 1.5 m high, pretensioned to 400 kN. The tangent stiffness at the origin is 35.00 kN/m³ and the curve barely bends: at the design load of 1 kN/m² the centre has moved 28.4 mm. What ends the story is not a stress. At 634 mm the hogging family's tension has fallen to zero and it goes slack, which happens at 25.21 kN/m² — 25.2 times the design load. Past that point half the net has stopped working and the rest has to find the whole load by sagging, so the real limit on a cable roof is a loss of geometry rather than a want of strength.01002003004005006000510152025deflection at the centre of the net (mm)load on the roof (kN/m²)the hogging family goes slack: 25.21 kN/m²25.2× the design loaddesign: 1 kN/m² at 28.4 mmk₀ = 35.00 kN/m³ at the origin
Fig. 10 What a prestressed system’s real limit is — not a stress anywhere, but a family of members going slack. A guyed mast is the two-member case of that, and its leeward guy is the member in question.

Where this goes next

00.20.40.60.811.2020406080100120140160midspan sag (m)total load on the cable (kN)the design load, 150 kNsolved 0.740 m1.125 mtangent here 341.2 kN/mk₀ = 8T₀/L = 133.3 kN/mthe flat-cable law The stiffness that comes from the shape A cable has no bending stiffness whatever, and it still holds up a roof. What resists the load is the change of its own geometry, so its stiffness is a function of the tension already in it — and prestress buys stiffness that no change of material could. 1968 kNthe restraint: 0.35 N/mm per mmtwo half-waves, each 6000 mmunrestrained 173 kN in one half-wave, drawn faintly · effective length 3555 mm Held everywhere, and it forgets its length A brace at a point divides a member's buckling length. A restraint spread along the whole member does something else — the member chooses its own number of half-waves, and past a few of them the critical load stops depending on the length at all. 00.20.40.60.80246810applied load ÷ buckling load1.1×1.3×1.4×2.0×3.3×10.0×first-order analysis says the answer is always 1×one over one minus the ratio The load that makes itself worse A structure that has leaned carries its weight off the axis, which makes it lean further. The amplification is one over one minus the load ratio, and it runs away long before the buckling load. 10⁻³10⁻²10⁻¹10¹00.20.40.60.81frequency (Hz)n·S(n), scaled to its own peakthe structure, 0.2 Hza stiffer one, 2 Hzbackground — the structure following the gusts: 70% of the turesonant — the structure ringing: 90% of the responsegust factor on the mean response: 4.39 The wind is a spectrum A wind load is quoted as a pressure, which suggests something steady. It is not — the energy is spread across four decades of frequency, almost all of it in gusts lasting minutes, and a tall building takes ninety per cent of its response from the sliver of that energy sitting at its own frequency.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

AnchorBeam columnBucklingCableElastic supportGeometric stiffnessGuyed mastNonlinearityPretensionSagSecond orderServiceabilitySlackTangent modulusWind load