Structural Engineering & Statics
where the load goes
Deflection

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.

Assumes The arch that gets shorter, The springings that make shortening worse and The settlement that matters is the difference.

17 min read 6 figures One support too manyThe load must go somewhereScale changes everything

Fixing an arch at its springings makes it six times as sensitive to its own shortening. A 60 m concrete rib at a 6 m rise, under 60 kN/m, loses 0.63 per cent of its 4,500 kN thrust when its springings are held, and the lost thrust acts at the elastic centre, 3.96 m up: 112 kN·m hogging at each springing, 58 sagging at the crown. Two-hinged, the same rib loses 0.11 per cent and carries 29 kN·m at its crown and nothing at its ends.

That calculation held the springings perfectly. An abutment is a block of concrete standing on ground, and under the springing moment and the thrust it does two things: it turns a little, and it slides a little outward. The question the earlier essay left was how stiff an abutment has to be before the arch behaves as fixed. It has an answer, and the answer turns out to be the less important half of what the abutment does.

A springing on a spring

Put each springing on a rotational spring of stiffness kθk_\theta — a moment per radian — and the fixed arch’s flexibility equation gains one term. The two redundants are the same as before, the change of thrust and the springing moment. The thrust’s coefficient is unchanged. The moment’s coefficient, the rib’s rotational flexibility ∫ds/EI\int ds/EI, gains 2/kθ2/k_\theta from the two springs.

With kθk_\theta infinite that is the fixed rib; with kθk_\theta zero, the springing moment has nothing to react against and vanishes, and the rib is two-hinged. Everything between is a rib whose springings are partly fixed.

The springing moment drains out through the abutment. The bending moment axial shortening leaves in a 60 m parabolic concrete rib at a 6 m rise (EI 420,000 kN·m²) under 60 kN/m, sagging upward, with each springing on a rotational spring. Fixed: −112 kN·m at the springings and 58 at the crown; kθ = 10 EI/L: −53 kN·m at the springings and 43 at the crown; kθ = 1 EI/L: −9 kN·m at the springings and 32 at the crown; two-hinged: 0 kN·m at the springings and 29 at the crown. Softening the springs takes moment off the springings and some off the crown, and the two-hinged rib is the limit with none at the ends.
Fig. 1 The moment shortening leaves in the 60 m rib (EI=420,000 kN⋅m2EI = 420{,}000\ \text{kN·m}^2, rise 6 m, 60 kN/m), sagging upward, with each springing on a rotational spring. Fixed: 112 kN·m hogging at the springings, 58 sagging at the crown. At a stiffness of ten times the rib’s EI/L: 53 and 43. At EI/L: 9 and 32. Two-hinged: nothing and 29.

The four curves are the rib at four springing stiffnesses. The springing moment drains away as the spring softens, and the crown moment drains with it — but only so far. At ten times the rib’s own EI/LEI/L the springings carry 53 kN·m and the crown 43. At EI/LEI/L itself, 9 and 32. The two-hinged limit, 29 at the crown, is the floor: whatever the abutment does, the crown keeps half of the fixed rib’s moment, because the thrust lost to shortening still acts on the crown’s lever arm when the springings are free.

The dimensionless group that measures the spring is κ=kθL/EI\kappa = k_\theta L/EI — the spring’s stiffness against the rib’s own rotational stiffness over its span. A spring of κ=1\kappa = 1 is as stiff as a span of the rib is in bending. It is the same comparison a column base makes against the column it carries, and the same ratio of spring to member decided whether a steel base plate on two bolts was fixed.

How stiff is fixed

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.
Fig. 2 The shortening moments as shares of the fixed rib’s, against each springing’s rotational stiffness as a multiple of EI/L, on a logarithmic scale. Solid, at the springings; dashed, at the crown. The springing moment is half the fixed value at 11.3 EI/L and nine tenths at 102; at EI/L itself it is 8 per cent. The crown never falls below the two-hinged rib’s 51 per cent.

On a logarithmic axis the springing moment rises along an S. It is negligible below κ=1\kappa = 1, rises through half the fixed value at 11.3, and reaches nine tenths at 102 and ninety-five per cent at 215. The crown moment rises over the same range from 51 per cent to all of it.

Those two numbers are the answer to the question as posed. An abutment counts as fixed — within ten per cent on the moment it is designed for — when its rotational stiffness is about a hundred times the rib’s EI/LEI/L. For this rib, EI/LEI/L is 7,000 kN·m per radian, so the requirement is about 700,000 kN·m per radian at each springing.

The S is wide: two decades of stiffness separate “nearly pinned” from “nearly fixed”. That is the shape that makes the question awkward in practice. Somewhere between a hundredth and a hundred times EI/LEI/L the answer is neither, and a design that assumes either end of the S for an abutment that sits in the middle of it has the springing moment wrong by a factor of two.

Where the lost thrust acts

The fixed rib’s redundants act at the elastic centre because that is the centroid of the rib’s flexibility. A spring adds flexibility at the springings, at height zero, and the centroid moves down to meet it.

Where the lost thrust acts, as the springings stiffen. The height above the springings at which the thrust lost to shortening effectively acts in a 60 m parabolic concrete rib at a 6 m rise (EI 420,000 kN·m²) under 60 kN/m, against the rotational stiffness of each springing as a multiple of EI/L, on a logarithmic scale. A rigid springing puts it at the elastic centre, 3.96 m up; at kθ = 102 EI/L it is 3.89 m, at 11.3 EI/L 3.38 m, at EI/L itself 1.34 m, and a pinned springing puts it on the springing line. Below that line the springings hog; above it the crown sags.
Fig. 3 The height above the springings at which the thrust lost to shortening effectively acts, against each springing’s rotational stiffness as a multiple of EI/L. Rigid springings put it at the elastic centre, 3.96 m up. At 102 EI/L it is 3.89 m; at 11.3, 3.38 m; at EI/L, 1.34 m. A pinned springing puts it on the springing line.

The line about which the lost thrust bends the rib is at 3.96 m when the springings are rigid — two thirds of the rise. At κ=102\kappa = 102 it has hardly moved, 3.89 m; at 11.3 it is 3.38; at 1 it is 1.34; and a pinned springing puts it on the springing line, where the two-hinged rib’s thrust acts.

That is the mechanism of the S in one picture. The springing moment is the lost thrust times the height of this line; the crown moment is the lost thrust times the rise minus it. As the springs soften, the line comes down, the springing’s lever arm shrinks to nothing, and the crown’s grows from 2.04 m to the whole 6 m — while the lost thrust itself shrinks from 28.4 kN to the two-hinged rib’s 4.9. The crown moment is the product of a growing arm and a shrinking force, which is why it falls only by half while the springing moment falls to zero.

Fixity belongs to the rib as much as to the ground

A rotational stiffness of 700,000 kN·m per radian is a property of a footing on ground, and the ground’s part of it is its shear modulus.

A rigid footing on an elastic half-space turns against a moment with a stiffness of 8GR3/3(1−ν)8GR^3/3(1-\nu), where RR is the radius of a circle with the same second moment of area as the footing’s base. For a footing 4 m along the span and 3 m across, RR is 2.1 m, and the stiffness runs from 180,000 kN·m per radian on soft clay (GG of 5 MPa) through 730,000 on stiff clay (20 MPa) and 2.2 million on dense sand (60 MPa) to tens of millions on rock.

Fixity is a ratio, and the rib is half of it. The springing moment from shortening as a share of the fixed value, against the ground's shear modulus on a logarithmic scale, for a 4.0 × 3.0 m footing turning alone: solid, a 60 m parabolic concrete rib at a 6 m rise (EI 420,000 kN·m²) under 60 kN/m; dashed, the same rib 10 times as stiff in bending. On stiff clay (G 20 MPa) the slender rib keeps 90 per cent of its fixed moment and the stiff one 49; on dense sand (60 MPa), 97 and 74. The same footing on the same ground is a fixed end for one rib and halfway to a hinge for the other.
Fig. 4 The springing moment from shortening as a share of the fixed value, against the ground’s shear modulus on a logarithmic scale, for a 4 × 3 m footing turning alone. Solid: the 60 m rib, EI=420 MN⋅m2EI = 420\ \text{MN·m}^2. Dashed: the same rib ten times as stiff in bending. On stiff clay the slender rib keeps 90 per cent of its fixed moment and the stiff one 49; on dense sand, 97 and 74.

For the slender rib, stiff clay already gives κ=104\kappa = 104, so the rib keeps 90 per cent of its fixed springing moment; dense sand gives 313 and 97 per cent. For practical purposes, this rib on this footing is fixed on anything firmer than soft clay.

Now make the rib ten times stiffer in bending — a deeper section, as a longer or more heavily loaded arch would need — and leave the footing alone. The same stiff clay now gives κ=10\kappa = 10, and the rib keeps 49 per cent of its fixed moment; dense sand gives 74. The same footing on the same ground is a fixed end for one rib and halfway to a hinge for the other.

That is what a ratio means. “Is this abutment fixed?” is not a question about the abutment. It is a question about the abutment against the rib, and a stiffer rib, which is exactly what a designer reaches for to control deflection and buckling, makes the same ground less able to fix it. The ground is a spring in every calculation it enters, and the question is always what it is a spring compared with.

Spreading is the same equation

An abutment does not only turn. The thrust pushes it outward, and on ground that is not rock it moves.

For a two-hinged arch, the earlier essay noted, a spread of the abutments is one more length change in the same equation as the rib’s shortening — and that remark is the whole of the matter for a fixed arch too. The rib’s shortening under its funicular thrust would, if nothing held the springings, close them by H0∫ds/EAH_0 \int ds/EA. For this rib that is 13.6 mm. An abutment that moves outward by 6.8 mm on each side opens exactly the gap the rib’s shortening closes, and adds exactly as much again.

A few millimetres at each abutment, and the shortening doubles. The fixed rib's springing moment (hogging, solid) and crown moment (sagging, dashed) in a 60 m parabolic concrete rib at a 6 m rise (EI 420,000 kN·m²) under 60 kN/m, against how far each abutment moves outward, rotation held. With no spread they are the shortening's 112 and 58 kN·m. The rib's own shortening under its 4,500 kN thrust would close the springings by 13.6 mm, so 6.8 mm at each abutment doubles both moments, and 10 mm each takes them to 278 and 143.
Fig. 5 The fixed rib’s springing moment (hogging, solid) and crown moment (sagging, dashed) against how far each abutment moves outward, rotation held. With no spread, the shortening’s 112 and 58 kN·m. The rib’s shortening under 4,500 kN would close the springings by 13.6 mm, so 6.8 mm at each abutment doubles both moments; 10 mm each takes them to 278 and 143.

The moments are linear in the spread and the shortening is simply the intercept. 6.8 mm at each abutment doubles every secondary moment in the arch; 10 mm each takes the springings to 278 kN·m and the crown to 143. Against a 60 m span, 6.8 mm is about a nine-thousandth.

The comparison with the rotation is lopsided in a way the question did not anticipate. Turning releases the springing moment, and the release is small until the abutment is soft by comparison with the rib. Spreading adds to both moments, and the addition is large as soon as the abutment moves by an amount comparable to the rib’s own shortening — which is a few millimetres, because a concrete rib carrying its design thrust is not very short.

What real ground does

A footing’s sliding stiffness on a half-space is 8GR/(2−ν)8GR/(2-\nu), with RR now the radius of a circle of the same area. The same 4 × 3 m footing, carrying 4,500 kN of thrust outward, moves 91 mm on soft clay, 24 on stiff clay, 8 on dense sand, 1 on weathered rock and a tenth of a millimetre on rock.

The ground decides by spreading, not by turning. The springing moment in a 60 m parabolic concrete rib at a 6 m rise (EI 420,000 kN·m²) under 60 kN/m standing on a 4.0 × 3.0 m footing at each end, on five grounds, against the fixed rib's 112 kN·m (dashed). Pale bars: the footing's rotation alone; dark bars: its rotation and its spread under the thrust. soft clay (G 5 MPa): kθ = 26 EI/L, 78 kN·m turning alone and 1,131 with 91.2 mm of spread; stiff clay (G 20 MPa): kθ = 104 EI/L, 101 kN·m turning alone and 457 with 23.8 mm of spread; dense sand (G 60 MPa): kθ = 313 EI/L, 108 kN·m turning alone and 237 with 8.0 mm of spread; weathered rock (G 500 MPa): kθ = 2,609 EI/L, 112 kN·m turning alone and 128 with 1.0 mm of spread; rock (G 5,000 MPa): kθ = 26,092 EI/L, 112 kN·m turning alone and 114 with 0.1 mm of spread.
Fig. 6 The springing moment in the 60 m rib standing on a 4 × 3 m footing at each end, on five grounds, against the fixed rib’s 112 kN·m (dashed). Pale bars: the footing’s rotation alone. Dark bars: its rotation and its spread under the thrust. Soft clay, 78 and 1,131 kN·m with 91 mm of spread; stiff clay, 101 and 457 with 24 mm; dense sand, 108 and 237 with 8 mm; weathered rock, 112 and 128; rock, 112 and 114.

Put both stiffnesses under the rib and the ground’s two effects separate cleanly. Rotation alone takes the springing moment from 112 kN·m to 108 on dense sand, 101 on stiff clay and 78 on soft clay: a relief, and a modest one. Rotation and spread together take it to 237 on dense sand, 457 on stiff clay and over 1,100 on soft clay. On weathered rock the two nearly agree, 128 against the fixed 112, and on rock they are indistinguishable from fixed.

The ground decides the arch by spreading, not by turning. The question of how stiff an abutment must be to count as fixed has a clean answer — a hundred times the rib’s EI/LEI/L — and on most ground the footing passes it easily. The question it does not ask, how stiff an abutment must be not to spread, has a much harsher one: the abutment has to move less than a few millimetres under the full thrust, and on any ground softer than rock a footing the size of this one does not.

The soft-clay bar should not be read as a design. A real abutment on such ground would be a much larger mass, or piled, or the arch would be tied — a tied arch holds its own span and its bearings are free, which is exactly the answer to everything on this page. The bar is there to show the direction and the scale: the elastic arithmetic is what explains why arches on soft ground are tied or are not built.

What a footing would have to be

The spread can be turned round into a requirement. Suppose the arch can tolerate secondary moments half as large again as the fixed rib’s shortening alone — 168 kN·m at the springings instead of 112. Then the two abutments together may open half of the 13.6 mm gap, 3.4 mm each, under the full 4,500 kN, and each abutment needs a sliding stiffness of about 1.3 million kN/m.

On dense sand that is a footing with an equivalent radius of 4.7 m: 69 square metres of base, against the 12 drawn above — nearly six times the area, for an arch whose springing moment then comes out at 168 kN·m, as asked. On stiff clay the same requirement is a radius of 14 m and 620 square metres, fifty times the footing; and asking for only a quarter more than the fixed rib’s moment quadruples both again. No arch of this size is founded that way. A footing’s sliding stiffness grows only with the first power of its size, so a stiffer abutment is bought by area, and the area runs away.

That is the arithmetic behind the habit long arches have of standing on rock. It is not that a fixed arch cannot be founded on soil — its rotation, as the earlier figures showed, is easily controlled. It is that a fixed arch on soil has to be designed for the moments its abutments’ spread puts into it, and those are set by the ground’s modulus and the thrust’s size in a way that more footing cannot economically change.

The older answer was a third hinge

There is one way to make the abutment’s movement irrelevant altogether, and it predates every calculation on this page. A three-hinged arch is statically determinate: with hinges at both springings and at the crown, its thrust follows from equilibrium alone, and a spread, a settlement, a temperature change or the rib’s own shortening simply moves it. No compatibility equation is written, so no imposed deformation produces a force.

The price is everything the fixity was bought for. The three-hinged rib is the most flexible of the three under live load and the most prone to buckling, its crown hinge is a detail that must carry the full thrust through a point, and the crown deflects under every load that passes. The two-hinged rib is in between: it is indifferent to rotation, but not to spread, which still enters its single compatibility equation over a denominator one sixth as stiff as the fixed rib’s.

So the three forms rank in exactly the opposite order for strength under load and for indifference to the ground, and the choice among them is a choice about the abutments as much as about the rib. Fixity is purchased with foundations, and on ground that spreads it is purchased at a price set by the ground.

What the elastic picture assumes

Every number on this page treats the ground as a linear elastic half-space and the thrust as applied once. Three things real abutments do sit outside that.

The ground’s modulus is not one number. The shear modulus that governs a footing under a sustained thrust is the drained, long-term one, and on clay it is several times smaller than the modulus that governs a short load. Rib shortening under dead load is a sustained effect, so the spreads above are, if anything, the short-term ones, and consolidation under a permanent thrust adds to them over years.

The rib creeps. Concrete under sustained compression shortens further with time, multiplying the axial term by roughly the creep coefficient. A rib whose instantaneous shortening is equivalent to 13.6 mm of spread is equivalent to two or three times that after a decade — which raises the shortening and reduces the importance of a given spread by the same factor, and does not change the conclusion that the two are added.

And the abutment turns and slides at once. A thrust applied above the footing’s base pushes it outward and tips it, and the coupling between the two motions is ignored here: each spring is independent. Coupling would make the abutment softer in both, and so make the spread larger and the rotation’s relief slightly larger too.

What the drawing does not include

It cannot show where the ground’s numbers come from. A shear modulus for a soil is a measurement with a scatter of a factor of two even on good sites, and the spreads above scale inversely with it. The figures show what a modulus does, not what any particular site’s modulus is.

It does not include the construction. An arch built on centring and struck after its abutments have taken their load may have spread before the rib carried anything — or may be built in two halves and closed at the crown after the abutments have moved, which removes most of the effect at the price of a closure detail. Which of those happened is a construction record, not a calculation.

And it treats the spread as uniform. An abutment on a slope, or with ground of different stiffness under its two edges, may move unequally, and an unequal settlement or spread between the two springings is antisymmetric — a different load case, which loads the line of thrust that must stay inside a masonry arch and bends a concrete rib in the mode that buckles it.

The arithmetic, by hand

Most of the essay is three numbers.

The rib’s shortening as a gap: H0∫ds/EAH_0 \int ds/EA. With H0=wL2/8f=4,500H_0 = wL^2/8f = 4{,}500 kN and EAEA of 20.4 million kN over a rib about 61.6 m long, that is 4,500×61.6/20,400,000=13.64{,}500 \times 61.6/20{,}400{,}000 = 13.6 mm.

The footing’s sliding stiffness on dense sand: a 4 × 3 m footing has an equivalent radius of 12/π=1.95\sqrt{12/\pi} = 1.95 m, and 8×60,000×1.95/1.7=552,0008 \times 60{,}000 \times 1.95/1.7 = 552{,}000 kN/m. Under 4,500 kN, plus the change the spread itself causes, it moves 8.0 mm, so the two abutments open 16 mm against the rib’s 13.6.

And the moments are linear in the total gap, so the springing moment on dense sand is 112×(13.6+16.0)/13.6112 \times (13.6 + 16.0)/13.6, less the few per cent the footing’s rotation takes back: 237 kN·m. An imposed deformation produces a force in proportion to the stiffness resisting it; here the deformation is the sum of two, and one of them was the ground’s.

What it comes to

Turning matters only when the abutment is soft compared with the rib. The springing moment is half its fixed value at a rotational stiffness of 11 times the rib’s EI/LEI/L and nine tenths at 102.

That is a ratio, so a stiffer rib is harder to fix. The same footing on stiff clay keeps 90 per cent of a slender rib’s springing moment and 49 per cent of one ten times as stiff.

Spreading adds where turning subtracts. The rib’s own shortening under its thrust is 13.6 mm of gap, so 6.8 mm of spread at each abutment doubles every secondary moment.

And on ordinary ground the spread wins. A 4 × 3 m footing on dense sand turns enough to release 4 kN·m and spreads enough to add 129.

Still open: the abutment that is built to move

A tie is one answer to a spreading abutment and a closure at the crown is another; a third is to let the abutment move on purpose and control where it ends. An arch whose springings sit on bearings that slide until a preset gap closes, or whose abutments are jacked outward before the crown is cast, starts its life with the spread already taken. Whether a jacked abutment can be set to cancel both the rib’s shortening and the ground’s long-term creep at once — or whether the two drift apart at different rates, so that a gap set right on the day of closure is wrong a decade later — is a question about two time-dependent movements that this page treated as one instantaneous number.

Where this goes next

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. 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. The force falls, the drift rises, and the damping goes the wrong way. What a compliant foundation does to a 0.6 s building on a 8 × 8 m footing, against the stiffness of the ground under it. Three curves, all normalised to the fixed-base answer. The period lengthens — 1.31 times at 200 m/s — because the swaying and rocking of the foundation are flexibilities in series with the structure's own, and the rocking term carries an h², so it is the tall building that feels it. The base shear falls with the period, which is why a fixed base is usually called conservative. The displacement rises, by 51% here, and that is what breaks the cladding, the services and the gap to the building next door. And the effective damping falls rather than rises: the structure's own is divided by the cube of the lengthening — 2.2% of an original 5% — while a slender building's foundation radiates only 0.28% back, because rocking radiates almost nothing at these frequencies. The ground is a spring Every dynamic result in this collection has assumed a structure rising from something that does not move. Nothing does. A foundation can slide and it can rock, both are flexibilities in series with the structure's own, and the rocking one carries a square of the height — so the period lengthens, the force falls, the drift rises, and the damping goes the wrong way.

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.

AbutmentArchAxial shorteningElastic centreImposed deformationPartial fixityRotational stiffnessShear modulusSoil-structure interactionSupport settlementThrust