Structural Engineering & Statics
where the load goes
Concept

Thrust — where it appears

The compressive force running along an arch, whose horizontal component is constant when the shape is the funicular of the load. It is wL squared over 8f for a parabolic arch under a uniform load, and what receives it - a foundation or a tie - is usually what decides the form.

Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.

The arch does not squash; it leans. The first two buckling modes of the same rib, drawn against its undeformed shape at an exaggeration of a few hundred. The first, at wcr L³/EI = 49.7, is antisymmetric: one half rises while the other falls and the crown moves sideways. The second, at 117.1, is symmetric — the whole rib settling. The two differ by a factor of 2.36, which is why an arch is braced against sideways movement of its crown rather than against the load it is carrying, and why a tied arch with a single hanger at midspan is doing nothing for the mode that governs it.

The arch that leans instead of squashing

A masonry arch is asked whether a line of thrust fits inside it. A steel rib is asked a different question entirely: it is a column carrying an axial force along its whole length, and the mode it buckles in puts one half up and the other half down while the crown moves sideways.

stability · Arch buckling The tie is a redundancy, so its stiffness decides the thrust. Thrust and rib bending for a 60 m tied arch of 0.15 rise ratio, against the stiffness of its tie. Cut the tie and the structure is a curved simply supported beam, so the tie force is the one redundant and the force method gives it: with a rigid tie the answer is 2227 kN, within 1.0 per cent of the funicular wL²/8f, and the shortfall is the arch's own axial shortening. A real tie stretches 68 mm and returns 2166 kN — 2.7 per cent of the flexibility is the tie — and whatever thrust the arch does not get, it carries as bending: 760 kNm at 30 m. A tenth of the tie stiffness is not a tenth of the problem; it is a different structure.

The thrust that never reaches the ground

The arches so far have all ended at the same sentence — the foundation is where an arch is really decided. A tie changes the sentence without changing the arithmetic: the horizontal force is still there, still the same size, and it now closes on itself through a bar at deck level.

structures · Tied arch The abutment force is the sag turned upside down. A 100 m ribbon carrying 35 kN/m at a sag of 2.0 per cent of its span. H = wL²/8f, so the horizontal force at each abutment is 21875 kN — 6.25 times the entire weight of the deck, and five times what a suspension bridge of the same span and weight at a tenth would have needed. The curve is a reciprocal and it has no flat part: halving the sag doubles the force, at any sag. What stops a designer flattening it further is not the ribbon, which is in tension and cannot buckle. It is what the ground at each end will take, and at 6.25 deck-weights that is usually rock or a very large anchor block.

The deck that is its own cable

Every other cable structure hangs something from the cable. A stressed ribbon hangs nothing — the walking surface is the catenary, laid at a fiftieth of the span rather than a tenth, because a footbridge has to be walkable. That one decision hands the abutments six and a quarter times the entire weight of the bridge.

structures · Stressed ribbon A tenth of a per cent of the thrust is all of the moment. The thrust a two-hinged arch loses to its own axial shortening, against rise-to-span. The flexibility equation's denominator has two terms — ∫y²ds/EI for bending and ∫cos²θ ds/EA for shortening — and their ratio is about (15/8)(i/f)², the square of the radius of gyration over the RISE. The loss is a little more than that ratio, because the released rib also shortens under its own shear, and at the 10 per cent rise drawn it is 0.11 per cent of the thrust, which sounds like a rounding error and is not: a parabolic arch under a uniform load is funicular, so the rigid solution has NO crown moment at all, and the 0.11 per cent that the rib shortening removes from the thrust leaves 29 kNm behind. The correction that is a tenth of a per cent of the thrust is a hundred per cent of the bending. At a two per cent rise the loss is 2.6 per cent, because a shallow arch's thrust is enormous and its lever arm is not.

The arch that gets shorter

A parabolic arch under a uniform load is funicular, so the perfect solution gives it no bending at all. Then the rib shortens under its own thrust by a tenth of a per cent, and every kilonewton-metre of moment the arch will ever carry comes from that.

deflection · Rib shortening Cross the hangers and the chords stop bending. The same tied arch, the same sixteen hangers, the same load on half the span — hung vertically and hung as a network. Vertical hangers make the two chords a Vierendeel frame, which has no truss action at all, so a partial load is carried by bending: 3316 kNm in the tie and 6234 in the arch. Inclined hangers can carry the shear between the chords axially, and the same load gives 686 and 831 — factors of 4.8 and 7.5. The thrust is identical in both, because that is decided by the span and the rise and nothing else.

Cross the hangers and the bending goes

A tied arch with vertical hangers is a Vierendeel frame with a curved top chord — it has no truss action at all, so a load on half the span is carried by bending. Incline the hangers so they cross and the same two chords become a truss.

structures · Network arch A load on one span moves the other. The change in level along two 100 m spans of stressed ribbon at a sag of 2.00 per cent, carrying 35 kN/m, when 20 kN/m of live load is added to the left-hand span, with a rigid pier (dashed) and with a pier of 200,000 kN/m (solid). With a rigid pier the loaded span sags a further 195 mm and the other does not move. With a pier of 200,000 kN/m the pier top leans 25 mm towards the load, the loaded span sags 358 mm and the unloaded span rises 191 mm; the pier carries 4,953 kN of the 9,440 kN a rigid pier would.

The pier that carries only the difference

A stressed ribbon continued over a pier pulls on it from both sides, and under its own weight the two pulls cancel. Load one span and they do not: the pier is asked for the difference, nearly ten meganewtons for a crowd on one of two hundred-metre spans. Whether it gives it depends on its stiffness against a number the ribbon supplies itself, and whatever the pier does not carry, the ribbon turns into movement — a third of a metre of extra sag in the loaded span and a fifth of a metre of rise in the empty one.

structures · Stressed ribbon The rib yields long before it would buckle. The largest combined stress, axial plus bending, anywhere in a 60 m two-pinned parabolic rib at a 12 m rise (EI 250,000 kN·m²) with a 12,000 mm² section 1,000 mm deep, against a factor on 15 kN/m of dead load over the span and 6 kN/m of live load over the left half together. Dashed: first order; solid: second order; dotted: the 355 N/mm² yield stress. First order, the rib reaches yield at a factor of 1.73; second order, at 1.18. The same load pattern, spread uniformly, would bifurcate the rib at a factor of 2.97, the right-hand edge — which is the number the eigenvalue reports and a number the rib never reaches.

The weight that bends a rib it cannot bend

A parabolic rib carries its own uniform dead load as pure thrust, with no bending in it at all. Put a live load on half the span and the rib bends in the shape of its own buckling mode from the first kilonewton, and the whole thrust, most of it from the dead load, multiplies that bending. The rib never reaches its buckling load. It yields well short of it, at a load its first-order check says it can carry.

stability · Arch buckling The same rib, fixed, bends at both ends as well as the crown. The bending moment that axial shortening leaves in a 60 m parabolic concrete rib at a 6 m rise (EI 420,000 kN·m², EA 20,400,000 kN) under 60 kN/m, sagging upward, from a frame solve of the rib. Dashed: two-hinged, sagging throughout, 29.4 kN·m at the crown. Solid: fixed at the springings, 57.9 kN·m sagging at the crown and 112.4 kN·m hogging at each springing. The funicular load itself leaves no moment in either rib; all of this is the 0.11 and 0.63 per cent of the thrust the two ribs lose by getting shorter.

The springings that make shortening worse

Fixing an arch at its springings is the stiffer, cheaper and usual way to build one in concrete, and it makes the arch six times as sensitive to its own shortening. The thrust it loses acts at the elastic centre, two thirds of the way up, so the moment lands at the springings as well as the crown, twice as large and the other way round — at the section the fixed arch is designed at, not away from it.

deflection · Rib shortening How stiff an abutment has to be to count as fixed. The shortening moments in a 60 m parabolic concrete rib at a 6 m rise (EI 420,000 kN·m²) under 60 kN/m as shares of the fixed rib's, against the rotational stiffness of each springing as a multiple of EI/L on a logarithmic scale: at the springings (solid, −112 kN·m when fixed) and the crown (dashed, 58 kN·m fixed, 29 two-hinged). The springing moment is half the fixed value at kθ = 11.3 EI/L and nine tenths at 102 EI/L; at EI/L itself it is 8 per cent. The crown moment never falls below the two-hinged rib's 51 per cent of the fixed value.

The abutment that spreads before it turns

A fixed arch is fixed only if its abutments hold still, and they have two ways not to. Turning releases the springing moment, but slowly: an abutment has to be a hundred times the rib's EI/L to hold nine tenths of it, and a footing on stiff clay under a slender concrete rib already is. Spreading does the opposite. The rib's own shortening is worth 13.6 mm of spread, so an abutment that gives 7 mm a side under the thrust doubles every secondary moment in the arch — and a footing on dense sand gives 8.

deflection · Rib shortening One span hung, the other still bare tendon. The level along two 100 m spans at a sag of 2.00 per cent, 35 kN/m finished and 3.0 kN/m of bare tendon, the tendons at 1100 N/mm² when finished, with the deck hung on the left span only (solid) and as finished (dashed), on a pier of 200,000 kN/m. The hung span sags 2.04 m and pulls 21,442 kN; the bare tendon beside it sags 0.20 m and already pulls 18,363 kN, because it was cut 455 mm shorter than the span so as to carry its share when finished. The pier is asked for the difference, 3,080 kN, and leans 15 mm towards the hung span — against 4,953 kN under 20 kN/m of crowd on one finished span.

The tendon that pulls before the deck arrives

A stressed ribbon is hung one span at a time, and the obvious fear is the stage at which one span carries its deck and the next does not: the pier between them asked for a whole span's thrust. It is not asked for that, because tendons cut to the finished length are already stretched across the empty span and pulling. For two 100 m spans at a fiftieth, the rigid pier's stage force is 4,109 kN against a crowd's 9,440 — unless the tendons were sized generously, when the stage overtakes the crowd.

structures · Stressed ribbon The water pushes on the chord, and the abutments push back along the arch. One horizontal ring of an arch dam, one metre high, 50 m below the water, spanning a 100 m gap with a central angle of 133.6°, drawn in plan with the water on the convex side. The water presses 491 kPa normal to the face everywhere (arrows); by the projection rule its resultant is that pressure times the chord, 49,050 kN per metre of height, pointing straight downstream. Each abutment pushes back along the ring's tangent, at 66.8° to the chord, so the two thrusts are N = pw/2sin(θ/2) = 26,686 kN — the pressure times the ring's radius, 54.4 m, which is the hoop formula of a pipe, arrived at without a pipe.

The angle that saves the most concrete

Cut an arch dam into horizontal rings and each ring is a curved surface with water on one side. The projection rule gives the water's whole push on it without an integral — the pressure times the chord — and the two abutments, pushing back along the ring, turn that into a thrust equal to the pressure times the radius, the hoop formula of a pipe arrived at without a pipe. From there one line of calculus says that the ring using least concrete has a central angle of 133.6°, whatever the canyon, the depth or the concrete. That number explains why arch dams in narrowing canyons are drawn at a constant angle rather than a constant radius, and why the angle that suits the concrete is not the one that suits the rock.

equilibrium · Curved surface pressure The arch is only as strong as what holds its ends. Load against mid-span deflection, as a share of the thickness, for a slab strip 4,000 mm long and 200 mm thick, a metre wide, with its ends held by springs of 0.05, 0.5 and 5 times its own axial stiffness EA/L and rigidly; dashed is its flexural load, 30.0 kN/m. At 0.05 EA/L it peaks at 45.3 kN/m, 1.51 times the flexural load, at 0.67 of its thickness; at 0.50 EA/L it peaks at 74.8 kN/m, 2.49 times the flexural load, at 0.45 of its thickness; at 5.00 EA/L it peaks at 108.6 kN/m, 3.62 times the flexural load, at 0.28 of its thickness; held rigidly it peaks at 116.8 kN/m, 3.89 times the flexural load, at 0.24 of its thickness. A soft restraint lets the ends move out as the strip tries to arch, so the thrust builds slowly and the peak arrives later and lower.

The edge that cannot hold the arch

A slab strip whose ends cannot move apart arches inside its own depth and carries nearly four times its flexural load. "Cannot move apart" is a stiffness, not a fact, and the arch needs a lot of it: half the gain takes end springs about half as stiff as the strip itself in compression, and nine tenths takes springs five times as stiff. An interior panel is held by the floor around it and gets the arch. An edge panel is held on its outer side by whatever runs along the edge, and a band of slab two metres wide between columns eight metres apart supplies a twentieth of what the strip needs. The arch is a property of where the panel is.

internal-forces · Membrane action Hung from its deck, the rib's bending is not amplified at all. The bending moment along a 60 m parabolic rib at a 12 m rise (EI 250,000 kN·m²) under 25 kN/m of dead load and 10 kN/m of live load on the left half. Standing free on pinned springings (solid) it peaks at 1,303 kN·m, against 576 at first order (dashed): the thrust amplifies the half-span bending 2.26 times. The same rib hung over a deck 0.10 times as stiff as the rib on vertical hangers (thick) peaks at 524 kN·m, 0.99 times its first-order 528, and the deck (thin) carries 59. The deck closes the thrust in tension, and the tension it carries straightens the deck as much as the thrust bends the rib.

The rib that cannot lean on its own tie

A free arch rib under a half-span live load bends in the shape of its own buckling mode, and its thrust amplifies the bending more than twofold. Hang a deck from the same rib on vertical hangers and close the thrust in the deck, and the amplification disappears: the thrust and the tie's pull move together, so the couple they make keeps its lever arm, and the only second-order term left is the small difference between the rib's deflection and the deck's. The critical load rises sixteenfold, and the deck's share of the bending is simply its share of the stiffness.

stability · Arch buckling The gaps that open after the arch is closed. For the 60 m fixed concrete rib rising 6 m under 60 kN/m, closed at 28 days with its abutments jacked apart to take up the day's gap: the gap between the rib and its springings that each movement opens after closure, in millimetres, against the years since, on a log scale — the rib's creep under its thrust, its shrinkage, and the ground consolidating under the thrust by 1.0 times its 8 mm immediate spread with 90% done at 10 years. After a hundred years they are 24.8, 16.2 and 8.0 mm, 48.9 in all, against the 13.6 mm the thrust shortened the rib on the day. The rib resists them while it creeps itself, so the gap it acts as if fixed against is 20.9 mm (fine dashes).

The jacks the arch forgets

Jack a fixed concrete arch's abutments apart before the crown is closed, by exactly the gap its thrust and its footings open on the day, and the springings start with no moment at all. A century later they carry 173 kN·m — half as much again as the 112 the jacks were set to remove. The thrust's own moment never relaxes; the jacks' correction does, as the rib creeps; and shrinkage and the ground's consolidation arrive after the jacks have gone. No single setting cancels movements that run on different clocks.

deflection · Rib shortening

Named alongside it

The objects these essays reach for when they reach for this one.

ArchAxial shorteningFunicularGeometric stiffnessAbutmentCable stiffnessHangerSecond-orderTied archAntisymmetric modeBucklingCreep

All concepts