Plane sections stay plane, and what the assumption costs

Every result about bending on this site descends from one sentence: a cross-section that was plane before the beam bent is still plane afterwards.

It is not a law and it is not derived from anything. It is a kinematic assumption — a statement about how the material moves — and it is what makes the strain proportional to the distance from the neutral axis, which makes the stress proportional too, which produces the second moment of area, the section modulus, the deflection formulae and everything built on them.

What it delivers #

Follow the consequences in order and it is striking how much rests on so little.

Bending a beam makes it curve. If sections stay plane and perpendicular to the axis, then two sections a small distance apart rotate relative to each other about some line, and every fibre’s change in length is proportional to its distance from that line. So

$$\varepsilon = \frac{y}{R},$$

with $R$ the radius of curvature. That is pure geometry — no material has been mentioned.

Add elasticity, $\sigma = E\varepsilon$, and the stress is linear in $y$ as well. Then the two equilibrium conditions on the cut face do the rest: the force sum puts the neutral axis at the centroid, and the moment sum produces $M = \sigma I / y$.

Notice what the assumption bought. Without it, the stress distribution over a cut face is an unknown function, and equilibrium supplies only two conditions — a force sum and a moment sum — which cannot determine a function. Plane sections reduces the unknown function to a single number, the curvature, and two conditions are then more than enough. The assumption is what converts an indeterminate problem into an arithmetic one, which is why it is the load-bearing sentence rather than a simplification of one.

Why it is true, when it is #

The justification is not that it is obviously so; it is Saint-Venant’s principle and a slenderness argument.

The real deformation of a loaded beam includes a shear distortion — sections do not stay perpendicular to the axis, because shear tilts them. What makes the assumption work is that for a slender member the bending deformation swamps the shear one.

The ratio scales with the square of the span-to-depth ratio: the bending deflection of a beam goes as $L^3$ or $L^4$ over $EI$, while the shear deflection goes as $L$ over $GA$. For a span-to-depth ratio of ten, shear contributes about one per cent of the deflection. At three, closer to ten per cent. At one, it dominates entirely and there is no meaningful sense in which the member is a beam.

The other half of the argument concerns distance from the load. Near a point load or a support the stress distribution is whatever the local geometry makes it, and only some distance away does it settle into the linear form. Saint-Venant’s principle puts that distance at roughly one section depth — which is why the three stress resultants that a cut reveals describe the state of stress accurately away from a load and not underneath one.

How it is known to be true #

An assumption that has never been tested is a hope, so it is worth saying how this one is checked, because the answer is unusually direct.

Strain is measurable. Bond a row of gauges up the side of a beam at one station, load it, and read them: the numbers should sit on a straight line, and the line’s intercept locates the neutral axis. That experiment has been done thousands of times since the 1930s and it is one of the first things done in any structural laboratory.

The result is consistent and specific. For a slender beam under pure bending, well away from loads, the gauges lie on a straight line to within the scatter of the instruments — a fraction of a per cent. The line’s zero falls at the centroid, as the force equation requires. And the departure grows exactly where the theory says it should: near loads, near supports, and as the member gets deeper.

That last point is what makes it a good assumption rather than a lucky one. It fails in a predicted place and by a predicted amount, which is the difference between a model with a known domain and a model that happens to work.

The four-point bending test is arranged specifically to interrogate it. Two loads symmetric on a simple span produce a middle region with constant moment and no shear at all — pure bending — so the region is free of the shear distortion the assumption ignores and free of any local disturbance. Theory is validated there rather than under a single central load, because a single central load puts the largest moment at exactly the station where the assumption is least valid.

Where it fails, and what fails with it #

Once the assumption goes, everything downstream goes with it, and the list is longer than it first appears.

Deep beams. A member with a span less than about twice its depth has a strain distribution that is markedly nonlinear, with the neutral axis displaced and the compression concentrated near the top. Its capacity is not $\sigma_y Z$ and its stiffness is not $EI$, and using either overestimates by a wide margin.

Regions near loads and supports. These are called disturbed regions, and they extend about one member depth from any discontinuity. Bearing zones, corbels, the region around a large web opening, the anchorage zone of a prestressing tendon, and the whole of a pile cap are all disturbed regions in which beam theory has no standing.

Sections with holes or abrupt changes. A step in depth, a notch or a large penetration invalidates the assumption locally, and the stress concentration that results can be several times the nominal value.

Composite and non-uniform members. The assumption is about geometry, so it survives a change of material — a composite section still has linear strain, with a stress that jumps where the modulus does. What it does not survive is slip between the components, and a layered member whose layers can slide has as many neutral axes as it has layers.

A member is a patchwork of two theories #

The boundary between where the assumption holds and where it does not runs through real members rather than between them, and the profession has a vocabulary for it.

A B region — B for Bernoulli — is a part of a member where plane sections holds and beam theory applies. A D region — D for disturbed, or discontinuity — is a part where it does not. Every member has both. The middle of a span is B; the metre or so around each support is D; the zone around a large web opening is D; the length either side of a concentrated load is D.

The rule of thumb for extent is the same one Saint-Venant’s principle gives: a D region reaches about one member depth from the discontinuity that caused it. So a beam of eight metres spanning at a depth of half a metre is B along nearly all of its length, and a beam of two metres at the same depth is D along most of it — the same member, the same section, and different theories in charge depending on how far apart the supports are.

The practical difficulty is that the boundary is a judgement, and both errors cost. Treating a D region as B overestimates its capacity, because beam theory assumes participation from material that is not participating. Treating a B region as D is safe and produces reinforcement layouts nobody wants to build.

Worse, the failures of a misidentified D region do not look like bending failures. They are bearing crushing, diagonal splitting, anchorage pull-out and punching — failure modes that a moment diagram cannot report on because they concern a distribution of stress the diagram replaced with three numbers. An engineer who checks the bending capacity of a corbel has answered a question that was never the one at issue.

The stronger version, in cracked concrete #

Reinforced concrete uses the assumption in a form considerably braver than steel does, and the bravery is usually invisible.

The tension side of a cracked concrete section carries nothing — the concrete has split and only the reinforcement crosses the crack. Nevertheless the analysis assumes the strain varies linearly across the full depth, crack and all, and uses that line to relate the steel’s strain to the concrete’s.

Taken literally that is nonsense. There is no material at the crack whose strain could be measured, and immediately either side of the crack the concrete is bonded to the bar and picks up tension again — a phenomenon called tension stiffening, which is why a cracked beam is stiffer than a fully cracked analysis says.

What the linear strain line actually describes is an average over a length spanning several cracks. Over that gauge length the deformation is smooth and the line is a good description; at any single station it is not. Beam theory has been quietly reinterpreted from a statement about a cut face to a statement about a smeared region, and it goes on producing correct answers because the quantity it feeds — the position of the neutral axis, and hence the lever arm — is an average quantity too.

There is a second bravery in the same calculation, and this one is licensed rather than merely tolerated.

That is the same manoeuvre as the smeared strain line, made explicit. The rectangle is not an approximation to the curve; it is a different shape chosen to match the curve in the only two integrals anything downstream will take. A triangle or a full rectangle matches neither and is nowhere near. Which is the honest statement of what plane sections is doing throughout: not describing the material, but supplying a distribution whose integrals are right.

That reinterpretation is the reason crack width has to be a separate calculation with its own empirical basis. The smeared model cannot produce a crack width, because it has smeared away the object being asked about. And it is why deflection in concrete is calculated with an effective stiffness interpolating between cracked and uncracked values rather than from either — the stiffness that matters is an average the section never actually has.

What replaces it #

For disturbed regions the profession does not patch beam theory; it changes method entirely, and the replacement is one of the more elegant things in the subject.

Strut-and-tie modelling treats the disturbed region as a truss. Invent a system of compression struts and tension ties inside the concrete, arranged to carry the applied loads to the supports; check the struts against the concrete’s crushing strength and the ties against the reinforcement provided; and if it works, the region works.

The justification is the lower-bound theorem: a force system in equilibrium with the loads that nowhere exceeds the material’s capacity proves the structure safe, without any claim to be what is actually happening. Since a truss is nothing but a force system in equilibrium, inventing one is exactly the manoeuvre the theorem licenses.

What makes it satisfying is that it puts the designer back in the position graphic statics occupied: choosing a load path rather than computing one. Two engineers can produce different strut-and-tie models for the same corbel, both valid, both safe, and requiring different reinforcement — and the better model is the one closer to the elastic stress field, because it demands less redistribution and therefore less cracking before it works.

What it really is: a reduction of dimension #

The assumption is usually presented as an approximation about how a section deforms. It is better read as a reduction of the problem’s dimension, and that reading explains why the subject is shaped the way it is.

A general elasticity problem carries fifteen unknown fields at every point of a three-dimensional body: six stresses, six strains and three displacements, connected by equilibrium, compatibility and a constitutive law. Solving it requires partial differential equations and, for anything but a handful of classical shapes, a computer.

Plane sections collapses that. If the strain across a cut is a straight line, then the entire deformation of a cross-section is described by two numbers — a reference strain and a curvature — and for pure bending by one. The body has gone from three dimensions to one: a member is now a line, its state at each point along it is a curvature, and the governing equation is an ordinary differential equation in a single variable.

Everything on this site that can be done by hand descends from that reduction. A moment diagram is a function of one variable. A deflection is a double integral along a line. A stiffness matrix has two rotations and two displacements per member end rather than a field. An influence line, a moment distribution, a collapse mechanism: each is arithmetic on a one-dimensional object, and each exists because a three-dimensional problem was reduced before any of them started.

Two numbers is a small container, and it is worth seeing what has to be thrown away to fit.

The stress in that third block exists because the section refused the shape it was offered. It is not a load effect, nothing applied it, and no free body will find it. And because the reduction keeps only the two numbers, it cannot distinguish the field that produced them from any other field with the same two.

Every quantity a beam analysis computes is identical for those two profiles, and one of them cracks a deck. That is the price of the reduction stated exactly: the two numbers it keeps are the two that equilibrium needs, and the stress that arises from the part it discarded is precisely the stress no equilibrium check can see.

Which is why the assumption cannot be patched when it fails. There is no small correction that recovers a linear strain distribution in a deep beam, because what has failed is not a number but the reduction itself — and the replacement, a truss invented inside the solid, is a different reduction rather than a repair of this one.

Where the model stops #

The assumption is the model. Every caveat elsewhere on this site about beam theory is, at bottom, this one. Slenderness, distance from loads, shear deflection and deep-beam behaviour are all the same statement seen from different angles.

Shear deformation, even when small. For a truss, web deformation can contribute a quarter of the total movement even though every member is slender, because the “web” is mostly air. Timoshenko beam theory adds a shear term and is the honest choice for deep or short members.

Torsion. An open section under torsion warps — the cross-section displaces out of its own plane, one flange forward and one back — so plane sections fails in a completely different way that has nothing to do with slenderness and everything to do with the shape of the section.

Slip between components. The assumption survives a change of material and does not survive a change of connection. A composite beam with full shear connection has one linear strain line across steel and concrete together; the same beam with partial connection has a step in the strain at the interface, and its stiffness and capacity both fall between the composite and non-composite values. The horizontal shear that has to be transferred is precisely what buys the single strain line, and a stack of planks with nothing between them is the limiting case where it has not been bought at all.

Large curvature. The strain expression assumes small deflections and small rotations. A member bent into a tight radius — a spring, a rolled plate, a cable over a saddle — needs a curved-beam theory in which the neutral axis is not at the centroid.

Local buckling. A thin plate that has rippled is no longer plane in the ordinary sense, so a class 4 section violates the assumption geometrically before it violates anything else.

The figures on this page carry a distortion worth naming precisely. The nonlinear strain curves are drawn from an interpolation chosen to have the right qualitative shape and the right limiting behaviour — they show that the distribution departs from the straight line and roughly how much, and they are not a solution of the elasticity problem for a deep beam. The straight line is exact within its assumption; the curve beside it is an illustration of a departure whose real form depends on the loading, the support conditions and the material.

The ladder from here #

Later rungs on this anchor: the strain distribution derived from the kinematics. Shear deformation and Timoshenko beam theory. Saint-Venant’s principle and its quantitative form. Deep beams and their strain fields. Disturbed regions and their identification. Strut-and-tie modelling in detail. Corbels, nibs and half-joints. Curved beams and the shift of the neutral axis. Warping torsion, where the assumption fails without any slenderness argument. And the finite-element method, which needs no such assumption and pays for its generality with an answer nobody can check by inspection.

Bernoulli proposed that plane sections remain plane in 1694, Euler used it to derive the elastica in 1744, and Navier put it together with elasticity into the bending formula in 1826. It is the oldest surviving assumption in structural engineering and the most productive, and it has been quietly wrong about deep beams for the whole of that time.