Structural Engineering & Statics
where the load goes
Equilibrium

The mechanism that stands up anyway

Two bars in a straight line between two pins are a mechanism by the count, by the rank and by every measure of how nearly a frame folds. Tension them and they carry load: 200 kN of tension gives the joint the stiffness of the same bars raised 1.8 degrees, and where the bars raised 3 degrees snap through at 11 kN, the tensioned flat pair carries 50 kN and goes on stiffening. The stiffness is a lack of fit of four millimetres, which 83 degrees of warmth takes away; compressed, the same pair buckles to rest crooked on one side or the other; and no tension can reach a mechanism that swings rather than shivers.

Assumes Counting the unknowns, and finding out whether statics can answer, The forces that are there with nothing applied and The count that does not see it.

18 min read 6 figures Geometry beats materialOne support too manyWhich failure arrives first

Rigid by every test, and still folding took two bars between two pins and raised the joint between them by a small angle. At 3 degrees the frame passed every test of the essays below it — as many unknowns as equations, a matrix of full rank, a condition number nobody would worry about — and was still nearly a mechanism: its own sag under a 50 kN load was most of its rise. At zero degrees the two bars lie in a line, and the frame is a mechanism by every test. The joint can move sideways, across the line, and to first order neither bar changes length.

That essay ended on the flat pair’s other property. The same two bars, pulled apart by their supports until they are in tension, resist a sideways push at the joint, because a joint moved sideways turns the tensioned bars and their tension acquires a component that pushes it back. A frame can be a mechanism by every count and still be a structure. The question was when a self-stress stiffens a mechanism and when it does not — and, since the flat pair is the limit of the frames the essay before had been measuring, how a mechanism that stands up compares with a structure that is nearly one.

Two bars in a line, pulled tight

The forces that are there with nothing applied found the flat pair’s self-stress state and the stiffness it lends the mechanism: each bar carrying tension TT over a length ll resists a sideways movement of its end with a stiffness T/lT/l, because a sideways movement does lengthen it — at second order, but the tension is there to be worked against. Two bars, 2T/l2T/l.

The bars here are the essay before’s: steel of 200 MN axial stiffness, 4 m each, across an 8 m span. Tensioned to 200 kN — 200 N/mm² in a bar of 1,000 mm² — the joint has a stiffness of 2×200,000/4,000=1002 \times 200{,}000/4{,}000 = 100 N/mm against a load across the line. That is not large. The same bars raised 3 degrees and unstressed have a stiffness of 2EAsin⁡2θ/l=2742EA\sin^2\theta/l = 274 N/mm, nearly three times as much.

The mechanism outcarries the frame that is nearly one. The load on the joint of two bars of 200 MN axial stiffness across 8.0 m against how far it moves down, followed to large displacement: the flat pair with its bars tensioned to 0, 50 and 200 kN with nothing applied, and the same bars raised 3 degrees with no tension, a rise of 210 mm. The raised frame is stiffer at first and snaps through at 11.1 kN, 89 mm down. The flat pair at 200 kN starts at 100 N/mm, carries 11.1 kN at 89 mm and 50 kN at 210 mm, and goes on stiffening; at 50 kN of tension, 50 kN at 242 mm; with none, at 252 mm.
Fig. 1 The load on the joint of two bars of 200 MN axial stiffness across 8 m against how far the joint moves down, followed to large displacement: flat, with the bars tensioned to 0, 50 and 200 kN, and raised 3 degrees with no tension, a rise of 210 mm (dashed). The raised frame snaps through at 11.1 kN, 89 mm down. The flat pair at 200 kN carries 11.1 kN at 89 mm and 50 kN at 210 mm; at 50 kN of tension, 50 kN at 242 mm; with none, at 252 mm.

Followed only a little way, the raised frame is the stiffer structure. Followed further, it is not a structure at all. As the joint comes down the raised bars flatten, their inclination falls, and the component of their compression that holds the joint up falls with it; at 89 mm, under 11.1 kN, the frame can take no more and snaps through to the far side of the line. That is the roof that jumps, in its simplest form. Its snap load has a closed form for a shallow pair, 0.385 EA θ30.385\,EA\,\theta^3, which for 3 degrees is 11.1 kN.

The flat pair goes the other way. As the joint comes down the bars turn away from the line, their inclination grows, and both their tension and the share of it that resists the load grow with it. The flat pair tensioned to 200 kN passes the raised frame at the raised frame’s own snap — 11.1 kN, 89 mm — and carries the 50 kN that the essay before applied to the raised frame at a movement of 210 mm, exactly the raised frame’s rise. There it is the raised frame turned upside down and in tension, and it keeps stiffening.

A mechanism by every test carries what a frame that passed every test cannot. The tests asked about the frame at rest. The tensioned flat pair has no stiffness at rest that the tests can see — its equilibrium matrix is rank-deficient and its elastic stiffness against the mechanism is zero — and it has a path that only rises. The raised frame has a stiffness the tests can see and a path that turns over.

What a tension is worth, counted as an angle

The two kinds of stiffness can be set side by side. A raised frame’s joint has a stiffness 2EAsin⁡2θ/l2EA\sin^2\theta/l, from the bars’ elastic resistance to stretching; a flat pair’s has 2T/l2T/l, from the tension. They are equal when

T=EAsin⁡2θ,T = EA\sin^2\theta,

so a tension can be read as the angle that would give the same stiffness without one.

What a tension is worth, as an angle. The tension that gives a flat pair of two bars of 200 MN axial stiffness across 8.0 m the same stiffness against a load at its joint as the unstressed pair raised by an angle θ: T = EA·sin²θ. Half a degree is worth 15.2 kN, one degree 61, three degrees 548 and six 2,185. A tension of 200 kN, which is 200 N/mm² in bars of 1,000 mm², is worth 1.81 degrees.
Fig. 2 The tension that gives the flat pair the stiffness of the same bars unstressed and raised by an angle θ, T = EA·sin²θ, for bars of 200 MN axial stiffness. Half a degree is worth 15.2 kN, one degree 61, three degrees 548 and six 2,185. A tension of 200 kN, 200 N/mm² in a bar of 1,000 mm², is worth 1.81 degrees.

The exchange rate is steep because the raised frame’s stiffness goes as the square of the angle. One degree of rise is worth 61 kN of tension; three degrees, 548; six degrees, over 2,000 — more than the bars could carry. So at small angles a modest tension does what raising the joint would do, and at the angles at which frames are usually built no tension in the bars could match the geometry.

That is the comparison the essay before was reaching for when it found the raised frame’s stiffness vanishing as the square of its angle. The road from a sound frame to its critical form ends at a mechanism, and a self-stress is a way of standing at the end of the road with some of the stiffness the road had lost. Two hundred kilonewtons of tension puts the flat pair back where the raised frame was at 1.8 degrees — at rest. Away from rest, as the first figure shows, it is in a better place than the raised frame ever was.

The tension matters most where the load is small

The three flat pairs in the first figure converge. At 50 kN the pair with no tension at all has moved 252 mm, the one at 50 kN of tension 242, the one at 200 kN 210. The tension that made all the difference near rest makes a sixth of it there.

The reason is that a flat pair’s stiffness has two parts, and only one of them is the self-stress. The other is the bars’ own stretch. A joint moved ww across the line lengthens each bar by about w2/2lw^2/2l, which adds EA w2/2l2EA\,w^2/2l^2 to its tension, and that added tension, turned through the bars’ slope w/lw/l, holds up a load of EA w3/l3EA\,w^3/l^3. An unstressed flat pair carries load by this term alone — nothing at first, then as the cube of the movement — and on these bars it reaches 50 kN at a movement of 252 mm, where 200×(252/4,000)3200 \times (252/4{,}000)^3 MN is 0.050 MN. A tensioned pair adds 2Tw/l2Tw/l, which is linear in the movement.

The two terms are equal where EA w3/l3=2Tw/lEA\,w^3/l^3 = 2Tw/l, that is at

w=l2T/EA,w = l\sqrt{2T/EA},

which for 200 kN in these bars is 4,0000.002=1794{,}000\sqrt{0.002} = 179 mm. Below that the tension carries most of the load; above it the bars’ own stretch does, and every flat pair, whatever it was tensioned to, follows the same cube. A self-stress is a stiffness for small loads. It decides how a flat pair behaves in the first few centimetres — whether it moves 89 mm under 11 kN or 135 — and hardly matters to how far a large load takes it.

The same distance appears once more, with the opposite sign. A flat pair compressed by TT buckles to the position where its bars have lost their compression, and that is where the stretch w2/2lw^2/2l has taken back the shortening Tl/EATl/EA: at w=l2T/EAw = l\sqrt{2T/EA} again. For 100 kN it is 126 mm, which is where the compressed pair of the next figure comes to rest. The crossover of the tensioned pair and the rest of the compressed one are one length, the movement at which the bars’ geometry has doubled or cancelled their self-stress.

The sign decides

A self-stress has a sign. The flat pair can be built a little long instead of a little short, so that the supports push its bars together instead of pulling them apart, and the same quadratic form that gave +2T/l+2T/l gives −2T/l-2T/l.

Tension stiffens the mechanism; compression buckles it. The load on the joint of two bars of 200 MN axial stiffness across 8.0 m, flat, against its movement either way: tensioned to 100 kN, unstressed, and compressed to 100 kN. Tensioned it starts at 50 N/mm and stiffens; unstressed it starts at nothing and stiffens as the cube of the movement; compressed it starts at −50 N/mm — it pushes the joint away — and the load it needs to hold the joint is most negative, −2.4 kN, at 74 mm. It rests, with nothing applied, at 127 mm either side, where its bars have lost their compression: a mechanism that stands, crooked, in one of two places.
Fig. 3 The load needed to hold the joint of the flat pair at a movement either way: tensioned to 100 kN, unstressed, and compressed to 100 kN. Tensioned, it starts at 50 N/mm and stiffens; unstressed, it starts at nothing and stiffens as the cube of the movement; compressed, it starts at −50 N/mm, needs its most negative load, −2.4 kN, at 74 mm, and rests with nothing applied at 127 mm either side.

A negative stiffness is not a softer structure. It pushes the joint away from the line as soon as it leaves it, so the flat position of a compressed pair is an equilibrium the joint leaves rather than returns to. The joint goes until the bars’ compression has been spent — until they have turned far enough off the line that their length is their natural length and they carry nothing. For 100 kN of compression in these bars that is 127 mm, either side, and there the pair rests, crooked and unstressed, with a positive stiffness again.

So a compressed flat pair is a bistable switch: two rests, 254 mm apart, separated by a barrier of 2.4 kN. It is the raised frame’s snap-through seen from the other end — the raised frame is a compressed pair that has already chosen a side and is waiting for a load to push it over — and the 2.4 kN is the load that does it from rest. A tension in the same bars has no such barrier anywhere along its path.

The stiffness is a few millimetres

Where does the tension come from? Nothing is applied to the flat pair. Its bars carry 200 kN because they were made shorter than the distance between the pins and stretched to fit: a lack of fit, which a frame built to the wrong length acquires by accident and a prestressed frame acquires on purpose. In a bar of 200 MN axial stiffness, 4 m long, 200 kN is a stretch of 4 mm.

A stiffness made of a few millimetres. The stiffness of a flat pair of two bars of 200 MN axial stiffness across 8.0 m against a load at its joint, against how much warmer its steel bars are than the ground that holds their ends, for tensions of 50, 100 and 200 kN. Each tension is a stretch of 1.0, 2.0 and 4.0 mm in a 4 m bar, and the bars lose it as they warm: the stiffness falls in proportion and is gone at 21, 42 and 83 degrees. An unstressed raised frame's stiffness hardly depends on its temperature at all.
Fig. 4 The flat pair’s stiffness at its joint against how much warmer its steel bars are than the ground that holds their ends, for tensions of 50, 100 and 200 kN — stretches of 1, 2 and 4 mm in each 4 m bar. The stiffness falls in proportion as the bars expand and is gone at 21, 42 and 83 degrees.

So the flat pair’s stiffness is a dimension of 4 mm, and anything that changes the dimension changes the stiffness. Warm steel bars by one degree against cold supports and each grows by 0.048 mm, which is 2.4 kN of tension in these bars. The 200 kN pair loses half its stiffness at 42 degrees and all of it at 83; a 50 kN pair, at 21 degrees. A tension that a winter’s day put in, a summer’s afternoon can take out, and a steel strand’s relaxation under a sustained strain takes out slowly whatever the weather does.

The raised frame has no such sensitivity. Its stiffness comes from its geometry, and warming its unstressed bars only raises its joint a little — a movement nobody applied, with no stress in it. A mechanism stiffened by a self-stress has traded a stiffness made of shape for one made of fit. It is a good trade where the fit can be kept — a cable net retensioned each year, a bicycle wheel trued — and a poor one where it is left to the weather.

Which mechanisms a tension can reach

The flat pair’s mechanism is a first-order one: the joint can start to move without any bar changing length, but it cannot go far without the bars stretching, at second order. That second-order stretch is what the tension works against, and it is why a tension can stiffen the mechanism at all.

Which mechanisms a tension can reach. Three pin-jointed frames, each with its one way of moving drawn dashed. Two bars in a line have one self-stress and one mechanism, and a tension in the bars stiffens the motion. Three bars in a line have one self-stress and two mechanisms, and the one tension stiffens both, because every motion of the two inner joints turns a tensioned bar. A parallelogram on two pins has a mechanism and no self-stress: its bars can carry nothing with nothing applied, so there is no tension to put into them, and its motion is not a first-order one that a tension could resist but a finite swing.
Fig. 5 Three pin-jointed frames with their one way of moving drawn dashed. Two bars in a line: one self-stress, one mechanism, and a tension stiffens the motion. Three bars in a line: one self-stress, two mechanisms, and the one tension stiffens both, because every motion of the inner joints turns a tensioned bar. A parallelogram on two pins: one mechanism and no self-stress, and a motion that is a finite swing rather than a first-order one.

Three bars in a line have two mechanisms — each inner joint can move across the line — and still one self-stress, the same tension in all three bars. The count, s−m=−1s - m = -1, says the frame is short of a member. It is, and the tension stiffens it anyway: every way the two inner joints can move turns at least one tensioned bar, so every motion is resisted. One state reaches every mechanism.

A parallelogram on two pins — two posts and a beam — is short of a member too, and nothing can be done for it. Its bars carry nothing with nothing applied, because there is no way to set them against each other: shorten the beam and the posts lean in, and the frame adjusts its shape instead of acquiring a force. It has no self-stress, and its mechanism is not a first-order shiver that turns into a stretch; the posts swing through any angle with every bar its own length. A tension stiffens only the mechanisms it can make the bars stretch to follow, and a mechanism that is a finite motion makes nothing stretch. That is the line between the frames a self-stress can rescue and the frames it cannot: the first-order mechanisms of a critical form, which the essay before measured the approach to, and the finite mechanisms of a frame with a member missing.

One tension and many shapes

A string is the flat pair continued. Eight bars across the same 8 m, every joint free to move across the line, have seven mechanisms and one self-stress, and the one tension stiffens all seven.

One tension stiffens every shape of a string. A string of eight bars across 8.0 m, every joint free to move across the line and the bars tensioned to 100 kN: seven mechanisms and one self-stress. The stiffness of each of its seven shapes, from one bulge to seven, computed from the tension alone (bars) and from (4T/a)·sin²(kπ/2n) (dots). The one-bulge shape has 15.2 N/mm per unit of its shape, the seven-bulge 384.8, 25 times as much. Every one of them is set by the tension and none by the bars' axial stiffness, which is why a string's pitch is a measure of its tension.
Fig. 6 A string of eight bars across 8 m, tensioned to 100 kN: the stiffness of each of its seven shapes, from one half-wave to seven, computed from the tension alone (bars) and from (4T/a)·sin²(kπ/2n) (dots). The one-half-wave shape has 15.2 N/mm, the seven-half-wave 384.8, twenty-five times as much.

Each shape has its own stiffness, from 15.2 N/mm for the single bulge to 384.8 for the shape that zigzags at every joint, and every one of them is set by the tension and the spacing alone: (4T/a)sin⁡2(kπ/2n)(4T/a)\sin^2(k\pi/2n), with nothing of the bars’ axial stiffness in it. That is why a string’s pitch tells its tension, and why the tension in a stay can be read off a frequency: a mechanism stiffened only by a self-stress vibrates at frequencies that are a measure of the self-stress, and of nothing else. A tensioned flat pair is a one-joint string, and its natural frequency at the joint is the square root of 2T/l2T/l over the mass there.

The same picture is the stiffness that comes from the shape of a cable, which is the string with infinitely many joints, and of a cable net, which is a string in two directions at once. In each, the structure is a mechanism of very many shivers, and one self-stress, set by a few millimetres of fit, is what makes it stand.

The flat pair by hand

Take the flat pair tensioned to 200 kN under 50 kN at the joint. Guess the joint moves 210 mm. Each bar is then 4,0002+2102=4,005.5\sqrt{4{,}000^2 + 210^2} = 4{,}005.5 mm long, 5.5 mm longer than at rest, which adds 200×5.5/4,000=0.275200 \times 5.5/4{,}000 = 0.275 MN to its tension: 475 kN. The vertical component of the two tensions is 2×475×210/4,005.5=49.82 \times 475 \times 210/4{,}005.5 = 49.8 kN, the load. The guess was right.

The raised frame’s snap: the bars start at 3 degrees, and a shallow pair snaps when the joint has come down by about 1−1/3=421 - 1/\sqrt 3 = 42 per cent of its rise, where the load is 0.385 EA θ3=0.385×200,000×0.05243=11.10.385\,EA\,\theta^3 = 0.385 \times 200{,}000 \times 0.0524^3 = 11.1 kN. The joint is then at 0.42×210=890.42 \times 210 = 89 mm. The flat pair at 89 mm: each bar is 1.0 mm longer, carrying 200+50=250200 + 50 = 250 kN, and the load is 2×250×89/4,001=11.12 \times 250 \times 89/4{,}001 = 11.1 kN. Where the raised frame gives up, the tensioned flat pair is carrying exactly the same.

Bars, pins and a fit that holds

The bars are pin-ended and straight, and the supports do not move. A support that yields under the pull of the tension lets the tension out exactly as warming does; a cable net’s edge beams and a wheel’s rim are part of the self-stress, and their flexibility is part of the fit. Bars that can bend would add a little stiffness of their own to the flat pair, and nothing to its sign.

Compression is allowed in the bars. The compressed flat pair is drawn as bars that can push; a pair of cables pushed together simply goes slack, which is a mechanism with no stiffness either side and no bistable rests. The flat pair’s tension is safe from that only while the load never takes it below zero, and the warm-day figure says how much margin a given tension leaves.

The analysis is static. A load applied suddenly to the raised frame near its snap load sends it over at less than the static value; the tensioned flat pair, whose path only rises, has no such threshold.

A mechanism is not a verdict

The count, the rank and the condition number answer the question of whether a frame’s equilibrium equations can be solved, and for the flat pair the answer is no. That is a statement about the frame at rest with nothing in it. It is not a statement about whether the frame can carry load. A first-order mechanism with a self-stress of the right sign carries load from the first millimetre, never snaps and stiffens as it goes; the frame a few degrees from it, which passes every test, carries more at first and snaps at a load set by the cube of its angle.

What the mechanism carries it carries on a fit of millimetres, and that is the price. The essays on determinacy asked whether a frame could stand; the honest answer, for the frames at the end of the road, is that some of them can stand only while something keeps them pulled tight.

Still open: the frame whose tension is set by its load

Every tension here was put in before the load and held by the supports. A cable roof, a guyed mast and a tensegrity tower all carry tensions that the load itself changes, sometimes by more than the prestress — and a mechanism whose stiffness is its tension then has a stiffness that depends on the load it is carrying, in both directions. A snow load that tightens one family of cables in a net slackens the other. Whether a prestressed net under a load that slackens some of its cables keeps every one of its mechanisms stiffened, or loses the ones those cables were holding, and how much prestress keeps every cable in tension under every load case, is the question the flat pair’s single cable cannot pose and every net must answer.

Where this goes next

Four ways of being nearly a mechanism. Four measures of the two-bar frame against the angle its bars make with the line between the supports, on logarithmic axes: the bar force as a multiple of the load, the condition number of the equilibrium matrix, the largest relative change of force from a one-millimetre error in any joint's position, and the joint's sag under the load as a share of the rise, for bars of axial stiffness 200 MN over a span of 8.0 m, loaded with 50 kN. The first three grow as one over the angle: at 2° the force is 14.6 times the load, the condition number 54, and a millimetre of error changes the force by 0.7 per cent. The sag grows as the cube: 4.3 per cent of the rise at 8°, 41 per cent at 4°, 3.1 times it at 2°. The measure that ends the analysis is not the arithmetic's. Rigid by every test, and still folding Counting the unknowns says whether a frame can be solved and the rank of its equations says whether it can stand, and both answers are yes or no. A frame a few degrees from a critical form passes both and is still nearly a mechanism — and of the ways that shows, the last to arrive is the one the rank test measures. Its own sag under load eats a tenth of its geometry at six degrees; sixteen-figure arithmetic would not notice anything until well below half a degree. Two null spaces of one matrix, and the count is their difference. Two pin-jointed frames, each with the forces it can carry with nothing applied to it drawn on its bars — tension one colour, compression the other, thickness in proportion. That force set is the null space of the equilibrium matrix; a mechanism is the null space of its transpose; and Maxwell's count b + r − 2j is the difference of their dimensions and knows neither of them separately. A square with both diagonals has s = 1 and m = 0. Two bars in a straight line has s = 1 and m = 1 with a count of 0, so the count is satisfied by a frame that both folds and can be prestressed — and the prestress stiffness is positive, which is why a tensioned pair of collinear bars is stiff at all. The forces that are there with nothing applied Maxwell's count is the difference between two dimensions, and it knows neither of them separately. A frame can satisfy it exactly and still both fold and be prestressable — and when it does, the second of those is what stops the first. The count is necessary and not sufficient. Two pin-jointed frames, each satisfying m + r = 2j exactly. One of them folds anyway, because the equations are not independent; the ghosted outline is the motion that costs no member any change of length, drawn at an exaggeration of 0.55 of the span. The count that does not see it A frame can have exactly as many unknowns as equations and fold up anyway. The count asks whether there are enough equations; it never asks whether they are different from one another.

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.

Critical formDeterminacyGeometric stiffnessLack of fitMechanismPrestressSelf-stressSnap-through