Four Point Eight Metres and a Rippling Cup
The extension runs 4.8 m clear, from the new steel over the old back wall to the cavity wall at the far end. Two hundred and forty-one millimetre I-joists at 400 centres, a 22 mm tongue-and-groove deck, plasterboard under. The calculation pack shows the joists at better than span over four hundred and eighty under imposed load, so the engineer signed it, the inspector passed it, and the floor sits comfortably inside every limit anyone applied to it. In week three the client rings to say the coffee on the island ripples when somebody walks past, and the oven door ticks against its catch.
Nobody made an arithmetic mistake. A deflection ratio and a felt bounce are two measurements of two different things, taken under two different loads, and satisfying one has never implied anything about the other. The ratio asks how far the floor sags when it is fully loaded, because sag is what cracks a plaster ceiling and opens a grout line. What the client is describing is how fast the floor moves under a load that weighs eighty kilos and is only there for half a second.
The number that answers the client is the floor's fundamental frequency — the rate the assembly wants to oscillate at once something disturbs it — together with how hard a footfall drives it and how quickly the motion dies away. Frequency is set almost entirely by span and depth, and it is fixed the afternoon somebody picks a joist off a chart. Everything after that afternoon is compensation, and compensation is always dearer than the extra 60 mm of depth would have been.
Three Criteria, One Joist, and They Do Not Agree
Take the deflection ratio first, because it is the one that gets checked. In the United States the binding values sit in IRC Table R301.7 Allowable Deflection of Structural Members and, outside the prescriptive residential route, IBC Table 1604.3 Deflection Limits; floors land at span over three hundred and sixty under live load, with a looser total-load figure alongside it. The Eurocode route reaches similar territory through national annex limits rather than one printed ratio. Every one of them is checked against the design imposed load, which is heavier than anything standing in the room on an ordinary Tuesday.
That load case is the whole difficulty. The resident judges the floor under permanent load plus one person, and permanent load is the only load that ever influences how the floor feels. The two criteria can even pull in opposite directions: a levelling screed laid to hit a flatness tolerance adds mass without adding much stiffness, and mass sits in the denominator of the frequency expression, which nudges the floor down towards the walking harmonics rather than away from them.
The third criterion is the one that matches what people actually do. Nobody assesses a floor by loading it uniformly. They stand in the middle of it and rock on their heels, which is a point load of roughly a kilonewton — about 225 lbf — applied at mid-span, and the deflection that produces is what the trade means by bounce. The residential floor clause of BS EN 1995-1-1, Eurocode 5, carries a limit on exactly that deflection, tightening as the span grows, with the National Annex supplying the values. CSA O86 in Canada goes further and publishes vibration-controlled spans for wood floors rather than leaving the question to a ratio at all. The IRC and IBC deflection tables say nothing about vibration, and were never written to.
| Criterion | Load it is checked under | What it actually protects |
|---|---|---|
| Live-load ratio, commonly span/360 | Full design imposed load, rarely present | Plaster, tile and rigid glazing — cracking, not comfort |
| Total-load ratio, commonly span/240 | Dead plus imposed, including long-term creep | Visible sag against a door head or a skirting line |
| Fundamental frequency | Permanent mass plus a modest occupancy allowance | Whether an ordinary walking pace can resonate the floor |
| Deflection under a 1 kN (225 lbf) point load | One person standing at mid-span | The bounce a client tests with their own heel |
| Peak acceleration against a comfort limit | A walker crossing the whole floor panel | What the complaint is describing when it arrives |
A Depth to Argue From
Before anyone opens a manufacturer's span table it helps to have a depth in the room, and the habit in residential work is a span-to-depth ratio around twenty to one. It is not a code value and nothing enforces it — it is where estimators start. On the 4.8 m span above, twenty to one lands near 240 mm, which is exactly why a 241 mm joist looked reasonable to everyone who glanced at it. Work at seventeen to one instead, the ratio you reach for when the floor has to feel solid rather than merely comply, and the same span asks for about 283 mm, which is a 302 mm joist. One rung up the depth ladder, and the depth ladder is a short list of discrete sizes, so it is one decision rather than a negotiation.
The ratio cannot know spacing, flange size, service load or whether the joists run continuous over an internal support, so it settles nothing on its own. What it does is force the depth conversation while depth is still free. Joist depth sets the floor zone, and the floor zone sets the threshold step, the head height under the new opening, the fall available for a wet room above and whether the new floor meets the old one flush. Finding out in week six that the floor wanted to be 60 mm deeper is not a structural problem any more, it is a demolition one.
Put the clear span through both ratios — twenty to one for the compliant answer, nearer seventeen for the solid one — and see whether the difference is a whole joist size before the schedule is priced.
The clear span the I-joist must cover.
The span-to-depth ratio used as a rough preliminary sizing target.
Minimum estimated joist depth
7.8 in
This is a rough preliminary rule-of-thumb only — always verify final joist depth against the manufacturer's official span/load table (e.g. TJI, LP, or similar), which accounts for actual load, spacing, and deflection limits.
They open the calculator with your figures already in it
I-Joist Minimum Depth Selector: 7.8 in — shown in imperial, US market. The link sets both, so the result they see is the one on your screen.
Add the equipment this sizes
This result is a specification — 7.8 in — not a quantity. Put the thing it sizes into your project: how many, what you call it, and your supplier’s price.
What this calculation does not cover
- A depth is not a product. Two joists of the same depth from the same maker can differ widely in what they carry, because flange size, flange material and web thickness change series by series — the lightest 241 mm (9.5 in) series and the heaviest are not interchangeable on the same span at the same spacing. Ordering by depth alone is how the right depth turns up in the wrong series.
- The ratio describes a joist between two supports; it says nothing about what carries those supports. The beam or header collecting the joist ends spans further and picks up a strip of floor, so it is nearly always the deeper member — set a floor-to-floor height from the joist depth here and the beam has nowhere to go, ending up dropped below the ceiling line or flush-framed with hangers. That is a detail decided early or paid for late.
Where the Frequency Comes From
For a uniform simply supported member the fundamental frequency is f₁ = (π/2)·√(EI / mL⁴), with m the mass carried per unit length. Span enters to the fourth power under a square root, so frequency falls with the square of the span and nothing else on the sheet competes with it. Stretching a floor from 4.2 m to 4.8 m — fourteen per cent more span, the kind of change a client asks for over the phone — costs about twenty-three per cent of the frequency at constant section and load. That is the single most expensive sentence anyone says during a design meeting.
Mass works against you, and it works against you slowly. Frequency varies with the inverse square root of mass, so doubling the load carried per metre only drops the frequency by about twenty-nine per cent. It is worth knowing which direction each material moves you: a topping adds mass and lowers frequency, while the stiffness it contributes through composite action is usually modest by comparison.
Depth is the lever that actually works. In an I-joist the flanges do nearly all the bending work, so the second moment of area rises roughly with the square of depth, and frequency — proportional to the square root of that — rises roughly in step with depth itself. Going from a 241 mm joist to a 302 mm one at the same span and load buys something in the order of a quarter more frequency. There is no other single change on the drawing that does that.
Walking is periodic, which is why frequency matters rather than stiffness alone. People walk at roughly 1.6 to 2.2 steps a second and each footfall carries harmonics at whole multiples of that rate, so excitation reaches up to about 8 or 9 Hz. AISC Design Guide 11, Vibrations of Steel-Framed Structural Systems Due to Human Activity, splits floors on that boundary: below it a walking harmonic can find the floor's own frequency and build a resonant response, above it each footfall arrives as a separate impulse and the floor never gets driven. Below about 3 Hz the first harmonic of ordinary walking lands directly on the floor, and no amount of damping makes that acceptable.
Frequency is not the acceptance test, and treating it as one is the common overreach. What a person feels is acceleration, and predicting that needs the effective weight of the vibrating floor panel, which comes out of the Design Guide's joist-and-girder procedure rather than from one member's properties. The guide's recommended peak acceleration limits are about half a per cent of gravity for offices and residences and around one and a half per cent for shopping malls, judged against the base curves in ISO 10137 and ISO 2631-2. SCI P354 in the UK and CCIP-016 from the Concrete Centre take the same idea through steel and concrete floors respectively.
Damping belongs in a sentence of its own so nothing implies it fixes the frequency. It does not move the frequency at all; it governs how large the response grows once resonance is reached. A bare structure with nothing on it sits near one or two per cent of critical, a fitted-out floor with ceilings, services and full-height partitions reaches three to five, and partitions are the biggest single contributor — which is why the open-plan extension is the hard case and why the same floor plan feels different after the joiner has been in.
Run the joist you are actually being offered against the mass that will really be sitting on it, and read which regime it falls in — resonant, low-frequency, or high enough that walking cannot drive it at all.
The clear span between supports, centre to centre of bearing.
The section's second moment of area about the bending axis, from its published properties.
The stiffness of the beam material.
The mass this beam carries along its length, including its own, at the loading present in service.
How quickly the floor's motion dies away, as a percentage of critical damping.
Fundamental natural frequency
5.94 Hz
This is a low-frequency floor, so a walking harmonic can resonate it and the response is governed by that resonance. Frequency alone does not settle whether it is acceptable — the full Design Guide 11 acceleration check, which needs the effective panel weight, does.
- Static deflection under the supported mass
- 0.35 in
- Dynamic amplification at resonance
- 16.67 (× static)
- Walking pace whose second harmonic matches this floor
- 178.23 steps/min
- Walking pace whose third harmonic matches this floor
- 118.82 steps/min
They open the calculator with your figures already in it
Floor Beam Vibration Natural Frequency Calculator: 5.94 Hz — shown in imperial, US market. The link sets both, so the result they see is the one on your screen.
Add the equipment this sizes
This result is a specification — 5.94 Hz — not a quantity. Put the thing it sizes into your project: how many, what you call it, and your supplier’s price.
What this calculation does not cover
- A single simply supported beam. A real floor vibrates as a panel of joists on girders, and the panel frequency is lower than any of its parts — Design Guide 11's combined-mode procedure exists for exactly that reason.
- Continuity, fixity and adjacent bays are not modelled. A continuous beam over several supports is stiffer and its frequency higher than this.
- Says nothing about acceleration, which is what people actually feel. Two floors at the same frequency can be worlds apart in comfort depending on effective panel weight and damping.
The Steel Over the Old Back Wall Moves Too
In an extension the joists almost never land on two solid walls. One end sits on a beam spanning the opening that was knocked through, and that beam deflects under the reaction the joists deliver into it. From the joist's point of view its support is moving, and the floor's frequency belongs to the system rather than to either member. Design Guide 11 handles this by adding the two static deflections before converting to a frequency, which is arithmetically the same as combining the two modal frequencies as 1/f² = 1/f_joist² + 1/f_beam².
Work an example and the result is sobering. A joist line at 8 Hz landing on a beam that is itself at 8 Hz gives a combined floor at about 5.7 Hz — squarely in the range a walking harmonic can drive. Good joists on a lively beam produce a lively floor, every time, and the beam is usually the cheaper of the two to fix: adding depth to one beam is a single line on an order, while adding depth to eighteen joists changes the whole floor zone.
Timber and engineered timber beams have a second problem the steel does not. They creep, so the deflection that was calculated on day one is not the deflection present in year five; Eurocode 5 accounts for it through the deformation factor it calls k_def, and the factor depends on the service class the beam ends up in. A trimmer over a bifold in a room that stays damp for a winter is in a different service class from one in a dry hallway, and the long-term sag differs accordingly.
Get the supporting beam's own deflection before you accept a joist frequency, because the two combine and the beam is the half that usually decides whether the floor lands above the walking harmonics or below them.
The uniformly distributed load along the beam's length.
The beam's clear span between supports.
The glulam's modulus of elasticity, from the manufacturer's grade stamp or design values.
The beam cross-section's moment of inertia about the bending axis.
The applicable code deflection limit, as a fraction of the span.
Calculated deflection
0.582 in
The deflection this beam works out to is below the L/360 limit for the span entered shown with it — you entered it from the deflection ratio you chose. Being under one limit is not a design. Nothing else is checked here — not the other limit states, not the connections, not the member the load arrives from.
- Allowable limit
- 0.65 in
They open the calculator with your figures already in it
Glulam Beam Deflection Checker: 0.5816 in — shown in imperial, US market. The link sets both, so the result they see is the one on your screen.
Add the equipment this sizes
This result is a specification — 0.582 in — not a quantity. Put the thing it sizes into your project: how many, what you call it, and your supplier’s price.
What this calculation does not cover
- The formula is the simply supported, uniformly loaded case only, Δ = 5wL⁴/(384EI). A cantilever of the same length and load deflects about nine and a half times as much, a beam continuous over a central support roughly two-fifths as much, and a central point load deflects a simple span about 1.6 times as much as the same total load spread evenly, so smearing a post or girder reaction into an equivalent uniform load understates the answer by around a third.
- This is the instantaneous elastic deflection. Timber creeps under sustained load, and the codes deal with it by multiplying the permanent share of the deflection before it is compared with the total-load limit, typically by about 1.5 in (38 mm) dry service and 2.0 when the timber is wet or green, with the exact factor jurisdictional.
- One deflection is compared against one denominator, but a floor is normally required to satisfy two separate checks: live load alone against the tighter limit and dead plus live against the looser one. Nothing on the page can tell which load has been typed in, so an L/360 pass on a total load is merely conservative while an L/240 pass on live load alone is not the floor check any code recognises. Meeting L/360 also says nothing about how a long-span floor feels underfoot, because perceived bounce is a vibration problem governed by frequency and mass rather than by this ratio.
- Only bending deflection is calculated. Shear deformation is left out, and in timber it is not trivial because the shear modulus is roughly one-sixteenth of E: on a deep beam with a span-to-depth ratio below about 15 it can add ten per cent or more. Whether your input already allows for it depends on whether the supplier publishes a true modulus or an apparent one that carries a shear allowance.
- The result is movement from the beam's unloaded shape, not sag below level. Glulam is routinely manufactured with camber, roof beams commonly to something like 1.5 times the dead-load deflection, so the installed beam finishes higher than this figure implies. On a shallow-pitch roof the same deflection is what starts ponding, where water collects in the sag and deepens it, and that feedback loop cannot be represented by a linear elastic formula.
- The moment of inertia is taken as a single constant for the whole span, so a tapered or pitched glulam, a notched end, or a section drilled for services is stiffer on this page than it is on site.
The Ends Do Half the Work
Every frequency on this page assumes the ends stay where they were drawn. On site they rarely do, and movement at a bearing adds directly to what happens at mid-span without appearing anywhere in the joist calculation. An I-joist bears through its bottom flange onto whatever is under it, and that flange is a small piece of engineered lumber taking a concentrated reaction: too short a bearing and the flange crushes, too little web support and the web buckles above it.
Manufacturers publish the numbers that govern this, not the code. Weyerhaeuser's Trus Joist TJI Joist Specifier's Guide and the equivalent literature for other products set minimum end and intermediate bearing lengths, state when a web stiffener is required, and give the squash block detail for a point load landing over a joist rather than beside one. That literature is tied to the product's own ICC-ES evaluation report, so a stiffener detail copied from one brand's guide is not evidence of anything about another's.
Hangers deserve their own look. A face-mount hanger has to have the right seat depth for the joist, the right fasteners in every hole the manufacturer nominates — a hanger with half its holes filled is not the hanger that was tested — and backer blocks where it lands on an I-joist header rather than on solid timber. A hanger also produces a pinned end, so any continuity you were counting on for stiffness quietly disappears at that support.
Continuity is worth chasing where the geometry allows it. A joist run continuous over an internal wall or a mid-span beam, rather than lapped and pinned at that line, is stiffer than two simple spans and its frequency is higher. Whether that is available depends on delivered joist lengths and on whether the depth zone will take a lap, both of which are ordering decisions, not site decisions.
Take the end reaction and the flange's allowable bearing stress and see how much bearing length it actually asks for — that comparison is what decides whether a web stiffener is on the drawing or missing from it.
The end reaction load the I-joist transfers into its bearing support.
The allowable compression stress perpendicular to the flange, from the manufacturer's design values.
The width of the I-joist's bottom flange, which is the bearing contact surface.
Required bearing length
1.77 in
A simplified proxy check — always follow the specific I-joist manufacturer's installation guide for actual web stiffener requirements at bearings and concentrated loads.
They open the calculator with your figures already in it
I-Joist Web Stiffener Bearing Length Calculator: 1.77 in — shown in imperial, US market. The link sets both, so the result they see is the one on your screen.
Add the equipment this sizes
This result is a specification — 1.77 in — not a quantity. Put the thing it sizes into your project: how many, what you call it, and your supplier’s price.
What this calculation does not cover
- Arithmetic can return a bearing shorter than any manufacturer permits. I-joist installation guides set an absolute minimum regardless of how small the reaction is — commonly 45 mm (1¾ in) at an end bearing and 89 mm (3½ in) where a joist runs over an interior support — so the 44 mm (1.7 in) these defaults produce is not a permitted detail. Round up to the published minimum before anything else is decided.
- The flange is only one half of the bearing, and usually the stronger half. The plate, beam or hanger seat underneath has its own compression capacity across the grain, and SPF at about 2.9 MPa (425 psi) crushes well before an LVL flange rated 3.5 MPa does — so on a softwood wall plate it is the plate that governs the bearing length, not the stress entered above. Check both and design to the lower.
- Bearing length says nothing about holding the joist upright. An I-joist end has to be restrained from rolling, by a rim board, blocking panels or a hanger with sides tall enough to grip the top flange — and where a post or a stud pack lands over the bearing from above, that load goes down through squash blocks fitted tight alongside the web, not through the joist. A web stiffener does neither of those jobs.
What Still Stiffens It After the Joists Are Ordered
A glued and screwed deck is the cheapest stiffness left on the job. Adhesive complying with ASTM D3498 Standard Specification for Adhesives for Field-Gluing Plywood to Lumber Framing for Floor Systems, laid in a continuous bead on a clean dry flange and closed up before the bead skins over, makes the deck and the joist act partly as one member and raises the effective second moment of area of the whole assembly. Screws hold it while the glue cures and stay put afterwards; nails withdraw over a heating season, which is how a floor that felt right at handover develops a squeak by the second winter. Panels run across the joists with the end joints staggered, and every panel edge either lands on a support or is tongued into its neighbour, because an unsupported edge is where a heel finds the difference.
Blocking, herringbone strutting and strongbacks do something the frequency calculation does not show. They contribute little to a single joist's own frequency, but they share a concentrated load across several joists, and a concentrated load at mid-span is precisely what the client applies when they test the floor. Lateral restraint at supports and bridging within the span are required by the wood floor framing provisions of the IRC once the joist proportions demand it, and open-web floor trusses carry a manufacturer requirement for a strongback through the web openings that is not optional and is routinely left out.
The ceiling underneath is a genuine trade-off rather than a free gain. Plasterboard screwed directly to the joists adds damping and a little composite stiffness. Hang it on resilient bars or clips for acoustic separation and that connection is deliberately broken, which is correct for airborne sound and costs you some of what the floor had. Both details are covered as workmanship by ASTM C840 Standard Specification for Application and Finishing of Gypsum Board, and neither is wrong — but decide it knowing the floor is a party to the decision.
Toppings cut both ways, and the honest answer depends on which side of the boundary the floor already sits. A gypsum or cementitious topping — Maxxon Gyp-Crete and USG Levelrock are the products usually named on a drawing — adds mass, adds damping and spreads a point load beautifully. On a high-frequency floor that is a clear improvement. On a floor already at five or six hertz, the mass drags the frequency further down towards the walking harmonics while the damping only reduces the amplitude of a resonance you did not want to have. Adding weight to a lively floor is not a reliable cure.
Partitions are the reason floors go quiet during fit-out. A full-height partition standing on the deck adds damping, and where it happens to sit over a joist line near mid-span it acts as a support and changes the span outright. That is also why an open-plan kitchen extension is the least forgiving case on the list: it has the longest spans, the fewest partitions, hard flooring with no carpet to absorb anything, and a client who spends every evening standing in the middle of it.
- Set the blocking, strutting or strongback rows before any deck goes down, because none of it can be fitted from above once the floor is closed.
- Check every joist end for bearing length, web stiffeners and hanger fasteners while the ends are still visible, and photograph the run.
- Sweep the flanges clean and dry — an adhesive bead laid onto sawdust is a bead that has already failed.
- Glue and lay only as much deck as can be screwed off before the bead skins, working across the joists with end joints staggered.
- Screw the deck rather than nailing it, at the panel manufacturer's edge and field spacing, and drive the heads flush rather than through the face.
- Walk the finished deck before the ceiling closes, and record what it feels like against the datum reading — a bare deck reads livelier than the finished floor will, and that difference is your evidence later.
What a floor is made of, from the walking surface down
- Finish and topping — adds mass and damping and spreads a point load, but the mass lowers the frequency — a help on a stiff floor and a hindrance on a lively one Self-Leveling Underlayment (SLU) Volume Calculator
- Structural deck — glued and screwed, it works partly as one member with the joist beneath and is the cheapest stiffness available after the joists are ordered Plywood and OSB Sheet Calculator (Subfloor, Wall and Roof)
- I-joists — the flanges do nearly all the bending work, so depth is the one change that moves the floor's frequency instead of merely its sag I-Joist Minimum Depth Selector
- Blocking and strongbacks — shares a concentrated load across several joists, which is exactly the load a client applies when they stand mid-span and rock on their heels
- Ceiling below — screwed direct it lends damping and a little stiffness; hung on resilient bars for sound separation, that contribution is deliberately given up Resilient Channel (RC-1) Spacing & Linear Footage Calculator
Testing It Before the Plasterer Arrives
The heel-drop is the field excitation the floor vibration literature is built on, and it takes ten seconds. Rise onto the balls of the feet at mid-span and drop onto the heels, then watch how the floor answers — a glass of water on the deck, a laser line thrown across the room from a tripod standing on the ground rather than on the floor being tested. What you are reading is not a number so much as a character: a single firm thud that stops, or a wobble that carries on for two or three cycles after the impact has finished.
The measurement that does produce a number is the point-load deflection, and it needs nothing beyond a dial gauge and a datum taken from below. Stand a person of known mass at mid-span, on the joist line and then between joists, and record the movement against a reference that is not part of the floor. That is the quantity Eurocode 5's residential clause limits and the quantity CSA O86 builds its vibration-controlled spans around, so there is something to compare the reading with rather than an opinion.
A phone accelerometer, laid flat on the deck and recorded through a heel-drop, will give a spectrum with the fundamental clearly visible, and that is a genuinely useful sanity check against the frequency you calculated. It is not an acceptance test. The comfort assessments in Design Guide 11, SCI P354 and CCIP-016 compare a predicted acceleration against a limit derived from ISO base curves, and a phone lying on an unfinished deck is measuring neither the finished floor's damping nor its final mass.
Record whatever you measure, with the date and the state of the build. A floor tested on bare deck, again after the ceiling, and again after partitions is three different floors, and knowing how much the fit-out actually contributed is what tells you whether the next extension of the same span needs one more joist size or does not. It is also the only defence available when a complaint arrives in year two about a floor that met every criterion it was ever checked against.
Settle these before the joist order leaves
Six things that are free while the schedule is still a spreadsheet and expensive once the joists are on a lorry, because every one of them changes the floor zone.
- Clear span, bearing to bearing — Frequency falls with the square of it, so 300 mm saved by moving a support is worth more than anything you can do to the floor afterwards.
- Joist depth, one rung above the ratio — Twenty to one gets a compliant floor; nearer seventeen gets a solid one. Depths come in a short discrete list, so it is one step, not a continuum.
- The mass that is genuinely there in service — Permanent load plus a modest occupancy allowance, with any topping or screed counted in — not the design imposed load the ratio was checked against.
- The beam at the open end, and its own deflection — Joist and beam frequencies combine, so a lively beam under good joists is a lively floor. Depth on one beam is cheaper than depth on eighteen joists.
- End conditions, in the manufacturer's own literature — Bearing length, web stiffeners, squash blocks, hanger fastener schedule, and whether continuity over an internal support is available at all.
- Ceiling fixing and topping, decided together — Direct-fixed plasterboard lends damping; resilient bars give it up for sound. A topping helps a stiff floor and can worsen a lively one.
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