The deflection that belongs to the support

A deflection calculation begins by drawing a beam with a triangle under each end. The triangle means the point does not move, and every result that follows is a statement about a beam whose ends are where they were.

Almost nothing is supported like that. A secondary beam sits on a primary beam, which bends. A primary beam sits on a column, which shortens. A column sits on a pad, which settles. A precast unit sits on a bearing, which compresses. Each of those is a spring, each is in series with the member above it, and a series combination is dominated by whichever is softer.

The arithmetic, on a floor #

Take an 8 m secondary beam of $EI = 84{,}000$ kNm², carrying 20 kN/m. On rigid supports it deflects

$$\delta = \frac{5wL^4}{384EI} = 12.7\;\text{mm}$$

which is span over 630 and comfortable.

Now put it on primary beams of the same section, spanning 8 m, receiving it at their mid-spans. A simply supported beam’s stiffness at mid-span is $48EI/L^3 = 7{,}875$ kN/m. The secondary delivers 80 kN to each, so each primary goes down by

$$\frac{80}{7{,}875} = 10.2\;\text{mm}$$

and the secondary’s mid-span, which sits on top of that, ends up at $12.7 + 10.2 = 22.9$ mm — span over 350, which is not comfortable and would fail most deflection limits for a floor supporting brittle finishes.

Forty-four per cent of that movement is not the secondary beam’s. It is the primary beam beneath it, doing exactly what it was designed to do.

The two halves of that number obey different laws, which is the first sign they are different quantities. The 12.7 mm follows span to the fourth: double the secondary’s span and its own deflection is multiplied by sixteen. The 10.2 mm does not follow it at all — it is a reaction divided by a stiffness, linear in the load, and the span it contains is the primary’s rather than the secondary’s. Lengthening the beam whose deflection is being checked changes one term by a factor of sixteen and the other by a factor of two.

Which free body produced the number #

The two contributions come from different free bodies, and keeping them apart is the whole method.

Cut the beam free of its supports and apply the reactions. The relative deflection of mid-span with respect to a straight line joining the two support points is $5wL^4/384EI$, and it is a property of the beam and its load alone.

Now take the support, with the reaction applied to it. It moves by $R/k$, and $k$ is a property of whatever is underneath — a beam, a column, a bearing pad, a pile.

The absolute deflection of any point on the beam is the sum: the movement of the line joining the supports, plus the beam’s own departure from it. The two do not interact, provided the structure is determinate — which is why the calculation is an addition rather than a solve.

The first of those two free bodies is what every standard method computes. One deflection without solving everything evaluates it as the integral of $M\bar{m}/EI$ along the member, and extending that integral to include the supports needs one extra term per support, $\bar{R}R/k$ — the same virtual-work statement with springs added to the list of things storing energy. Nothing about the method resists the extension; it is simply that nobody writes the extra term down, because the support does not appear on the sheet the beam is being sized on.

Two flexibilities in series #

Write the two contributions as flexibilities — deflection per unit load — and the shape of the answer is fixed:

$$\delta_{total} = \left(f_{beam} + f_{support}\right) W$$

Flexibilities add. Stiffnesses do not; they combine as $1/k = 1/k_1 + 1/k_2$, which is the harmonic sum this collection meets whenever two mechanisms act one after the other along a load path.

Three consequences follow, and each answers a question a designer actually asks.

The softer flexibility governs, and improving the other one is nearly wasted. Doubling the secondary beam’s second moment halves its own 12.7 mm to 6.4 and leaves the 10.2 alone: 22.9 becomes 16.5, a 28% improvement for twice the steel.

There is no point making either much stiffer than the other. The table below is the whole design guidance on the subject:

The two can be swapped for one another at a fixed rate. A support twice as stiff is worth exactly as much as a beam twice as stiff, when the two are equal — and worth twice as much when the support is the softer half. That exchange rate is the only design information a series combination contains, and it is available before either member has been sized.

And the limit has to be applied to the total. A deflection limit of span over 360 is about what a finish can tolerate, and the finish does not know which part of the movement belonged to which member. Checking the secondary beam alone against its own limit and the primary alone against its own is a check that both pass and the floor fails — a mistake that is easy to make because each member has its own calculation sheet.

What the limit is measured between #

The section above ended by saying the limit has to be applied to the total, because the finish does not know which member the movement belonged to. That is right for some finishes and wrong for others, and separating the two is worth doing because it decides which of the two numbers a designer should be worrying about.

A deflection limit is a proxy for damage, and damage is caused by relative movement between the two points something spans between. A movement that carries a finish and both its ends down together does nothing to it at all. So the question is never “how far did the floor drop” but “how far did these two points move relative to each other”, and the answer depends on where the two points are.

Work through the same bay.

A partition running along the secondary beam, from one primary to the other, sees the secondary’s own 12.7 mm and none of the primary’s. Its two ends sit on the primaries and go down 10.2 mm with them; the partition is translated bodily and its sag relative to its own ends is the beam’s contribution alone.

A partition running across the bay, from a column to a point above the middle of a primary, sees the primary’s 10.2 mm and none of the secondary’s — because it is spanning between a point that does not move and a point that moves by the primary’s deflection.

A ceiling or a screed spanning the full bay diagonally, from column to column, sees the whole 22.9 mm, which is the case the earlier section described.

And perimeter cladding sees only the movement of the edge beam it is fixed to, which may be neither.

So there is no single deflection of a floor. There is a field of vertical movement over the plan, and a serviceability limit is a statement about the second difference of that field along one particular line — with a different answer for every direction a finish might run.

Two consequences follow and they cut in opposite directions, which is why the member-by-member habit survives at all.

The member check under-estimates. A secondary beam passing its own $L/360$ at 12.7 mm is carrying a ceiling that is actually experiencing 22.9. That is the failure the earlier section warned about and it is real.

And the member check over-estimates. Applying the 22.9 mm total to a partition that runs along the beam is charging it for a rigid-body movement it never experienced. A floor condemned on a total that no finish in it actually sees has been condemned by an arithmetic convenience.

The specification that avoids both is one that states a relative deflection between named points — the drop of the middle of the bay relative to its corners, say — rather than a limit on any member. That is how the more careful cladding and partition specifications are written, and it is why a facade contractor asks for movements at fixing points rather than for a span-over-something.

There is a third case that belongs with the other two and is the one most often missed. A finish fixed to two different structures — a partition running from a floor slab up to the underside of the floor above, a stair landing spanning between a core and a floor plate, a lift door frame fixed to both the shaft and the slab — sees the difference between two independent movement fields, and neither member’s own check contains it. That is the same incompatibility a core and its columns produce at building scale, arriving one bay at a time.

The honest reason member limits persist is that they are cheap, they are checkable by the person who sized the member, and a floor that passes all of them is usually acceptable. The failures that do occur cluster exactly where this section predicts: at a finish whose span crosses the contributions of two different members, fixed to points that were checked separately by two people who each got the right answer to their own question.

In a continuous beam it moves the forces too #

Everything above is determinate, so the support flexibility changes deflections and nothing else. Make the beam continuous and it changes the forces as well.

That last sentence is the whole difference between the two problems. A settlement is a known movement, imposed. A flexible support is a movement proportional to the reaction, and the reaction is proportional to the stiffness — so the two are coupled and the answer requires solving them together.

Three equally stiff supports under a uniform load, with the middle one attracting the most reaction:

The middle support is relieved from 62.5% of the load to 35.9%, and the span moments rise by 64% to compensate. Which is exactly the redistribution nobody chose, arriving through the supports instead of through the joints.

There is a cousin of all this at the ends of the member rather than under them. A joint of finite rotational stiffness delivers only part of the fixed-end moment, and support flexibility and joint flexibility are the same phenomenon in two different freedoms — one a translation, one a rotation, and both springs the analysis usually replaces with a certainty.

The continuous version of the same idea #

The two problems are the same one at two densities. A Winkler beam has a modulus of subgrade reaction $k$ per unit length and a characteristic length $\sqrt[4]{4EI/k}$ that decides how far a load reaches. A beam on discrete springs has a stiffness $k$ at each support and a dimensionless ratio that decides how the load is shared. In both, the answer is a ratio of two stiffnesses and neither absolute value matters.

Where else it decides something #

Sideways. The floor plate distributing a lateral load to its walls is this problem turned on its side: the plate is the beam, the walls are the springs, and the ratio of the two flexibilities decides whether the load is shared by stiffness or by tributary span. Every conclusion on this page transfers to it with the word “deflection” replaced by “drift”.

Bearings. An elastomeric bearing under a bridge girder compresses by a millimetre or two under load and is deliberately soft in shear. Its vertical stiffness is part of the girder’s deflection and its rotational flexibility is part of the girder’s end restraint.

Piles and pads. A column on a pile group settles, and a frame on piles of different lengths settles differentially. In a rigid frame that is a redistribution, and it is one of the few cases where a geotechnical stiffness ends up in a bending moment.

Transfer structures. A column landing on a transfer beam has a support whose stiffness is the beam’s, which is orders of magnitude softer than the ground. Every column above it is on a spring, and the loads redistribute up the building — a column that would have taken its share by area takes it by stiffness instead, and the one over the middle of the transfer beam takes least of all.

And a column that shortens. In a tall building the column beneath a floor is getting shorter under everything built above it, which is a support movement arriving on a schedule rather than under a load. It is the same term in the same sum, with a creep coefficient in place of a stiffness.

Which of the two limits it threatens depends on the determinacy. Support flexibility is almost entirely a serviceability matter in a determinate structure and a strength matter in an indeterminate one, so whether it can be ignored is a question about the frame rather than about the member — and stiffness is not strength is the distinction it is trading on.

The rule that comes out of it #

Two flexibilities in series produce a design rule short enough to carry, and it is the opposite of the one intuition supplies.

Never improve the stiffer half. If the beam is contributing 56% of the deflection and its support 44%, the two are already well matched and neither improvement is efficient — every 1% of improvement to either buys about half a per cent overall. If the beam is contributing 14% and the support 86%, work on the support and leave the beam alone entirely; doubling the beam’s stiffness in that state improves the floor by seven per cent.

And the matched case is the expensive one to improve. That is the awkward corollary. A series combination whose two terms are equal is at its worst for marginal effort — there is no soft term to fix — and the only route left is to change the arrangement: shorten the primary’s span, add a column, or turn the two-way grid into a one-way system so that there is no primary beam under the secondary at all.

Which is why the answer to a floor at span over 350 is so often a layout change rather than a section change. The section change is available and is being asked to fix less than half the problem.

Where the model stops #

A support is not a linear spring. A bearing pad stiffens as it compresses; soil stiffness depends on the stress level and on the loading history; a bolted connection has slack before it has stiffness. A single number $k$ is a linearisation about an operating point.

The support’s stiffness depends on what else is on it. The 7,875 kN/m used above is the primary beam’s stiffness at mid-span with nothing else applied. Load the primary elsewhere and the secondary’s support moves for reasons that have nothing to do with the secondary — which is why a floor grid is really one structure and its members are analysed as though they were several.

Torsional and rotational flexibility have been ignored. A support that rotates as well as translating changes the beam’s end conditions, and for a continuous beam that is a larger effect than the translation.

And the two contributions have been added. That is exact for a determinate beam and approximate for anything else: in a continuous beam the support movements change the reactions, which change the support movements, and the sum is the result of a solve rather than an addition.

Where the number comes from, and why nobody has it #

The whole of this page is a ratio of two flexibilities, and one of the two is routinely unknown to within a factor of two.

A beam’s own flexibility is $L^3/48EI$ or thereabouts, and every term in it is on a drawing. A support’s is not. It depends on what the support is made of, on what else is bearing on it, on whether a bolt has taken up its clearance, on how a pad was bedded and on whether a pile has been loaded before.

The consequence is that support flexibility is very often handled by assuming one of the two limits — rigid, or a nominal spring — and checking whether the answer changes. That is a reasonable procedure and it has a specific failure: the two limits give the same forces in a determinate structure and very different ones in an indeterminate one, so the sensitivity check passes on the member and fails on the frame.

The honest position is that a beam’s own deflection is a calculation and its support’s is an estimate, and adding a calculation to an estimate produces an estimate. Which is an argument for stating the split — 12.7 mm of beam plus 10.2 mm of support — rather than quoting a total of 22.9 that reads as though it were known to three figures.

One thing does survive the uncertainty. Maxwell’s reciprocal theorem holds for the whole assembly, springs included: the deflection at the beam’s mid-span due to a load on the support equals the deflection at the support due to the same load at mid-span. Neither number is known to better than a factor of two, and the equality between them is exact anyway — a relationship between two unknown quantities is still a relationship, and it is the one check available on a model whose inputs cannot be verified.

What the pictures cannot show #

Every deflected shape here is drawn at an exaggeration of two or three orders of magnitude. The 22.9 mm this page is about, on an 8 m beam, is a slope of about one in 700, and a figure drawn to scale would show a straight line.

The sharing curve is drawn against a support stiffness with no units on the axis worth reading, because the quantity that matters is a ratio. A support stiffness of 7,875 kN/m means nothing without the beam it carries.

And nothing here shows what the occupant experiences, which is not a deflection but a change in one — a floor that moves when somebody walks across it, a door that binds when the bay next to it is loaded. Those are differences between two states, and every figure on this page shows one state.

The ladder from here #

Later rungs on this anchor: the grillage, where primary and secondary beams are solved together and the distinction between beam and support disappears. Rotational support flexibility, and the end restraint a real bearing supplies. Bearing design as a stiffness problem rather than a strength one. Piled foundations as springs, and the interaction between a raft, its piles and the frame above. Support stiffness in dynamics, where a soft support lowers the natural frequency of everything on it — which is how a floor becomes strong and unusable. And the measurement question: support stiffnesses are the least well known numbers in any model, and every result on this page is a ratio of two of them.