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.

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.

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.

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.

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 #

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.

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.

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.