Summary: Every document that governs a pile writes its verticality tolerance the same way, as a ratio of the deviation to the pile’s own length, and that choice changes what kind of quantity it is. The ladder runs from 0.4 percent, which is 1 in 250 or 0.229 deg, to 4 percent, which is 1 in 25 or 2.291 deg, a factor of ten, with ordinary work at 1 percent, 1 in 100 or 0.573 deg, for a bored pile and 0.5 percent for a precast driven one. Because the denominator is the pile itself, the permitted toe offset is the taper times the depth, which is 100 mm at 10 m, 300 mm at 30 m and 600 mm at 60 m at one percent. The same acceptance standard already publishes the integral, because its position tolerance is 70 mm plus 0.01 times the depth, and subtracting the taper leaves exactly 70 mm at every depth. A channel bolted at the cut off cannot see any of that, and the tolerance turns out to equal the apparent tilt from one hundredth of a g.

 

1. The Ratio That Sets Its Own Limit

Every document in the world that governs a pile writes its verticality tolerance in the same form, and none of them writes it as a length. A length would need a reference, and a pile has no fixed reference length. One foundation can hold a twelve metre pile next to a forty metre one, and nothing about the shorter pile justifies allowing it less lean. So the documents write a ratio of the deviation to the pile’s own length. The convention looks like an administrative convenience. It is a change of kind, and most of what follows comes out of it.

Here is the ladder, with every published figure converted into the forms a reader might need. The first column is the document and the pile type; the second is the figure as printed; the third and fourth are the same figure in the two forms a drawing uses; the fifth is what a channel reports.

Document and pile type	Ratio as printed	As 1 in	Millimetres per metre	Degrees
ICE, diaphragm wall with a reverse circulation mill	0.4 percent	1 in 250	4.0	0.229
Chinese acceptance standard, precast driven pile	0.5 percent	1 in 200	5.0	0.287
ICE, diaphragm wall with a hydraulic grab	0.7 percent	1 in 143	7.0	0.401
Chinese acceptance standard, bored pile	1.0 percent	1 in 100	10.0	0.573
ICE bored pile, and BS 8004 at 1 in 75	1.33 percent	1 in 75	13.3	0.764
AASHTO in soil, and IS 2911	1.5 percent	1 in 67	15.0	0.859
EN 1536, AASHTO in rock, FHWA at a quarter inch per foot	2.0 percent	1 in 50	20.0	1.146
French NF P94-262, where the technology is not fixed	3.0 percent	1 in 33	30.0	1.718
EN 12699, displacement pile	4.0 percent	1 in 25	40.0	2.291

The spread from end to end is a factor of ten, and the two ends describe the same physical object: a vertical pile in the ground carrying a building. There is no difference in the concrete, the steel or the load path between the top and the bottom of that table. There is a difference in what each document decided a contractor could deliver, and that decision is the whole content of the number.

Read as an angle, the upper end of the ladder is not a tolerance at all. Two percent is 1 in 50, which is a draft angle on a casting. Four percent, which the European standard for displacement piles permits by default, is 1 in 25, and a pile leaning 1 in 25 is a wedge rather than a column. A reader who has only ever seen the figure as a percentage has no feel for that, and the conversion is the first thing worth doing.

But the more useful observation is about the shape of the rule rather than its size. A tolerance written as a fraction of the object is a shape tolerance, and the shape it describes is a taper. It does not say the pile may lean by this much. It says the pile may deviate by this fraction of however long it happens to be. Those two statements coincide only on a pile of one particular length, and there is no such length in a real foundation.

2. What a Taper Tolerates at Depth

The consequence of a taper is easy to state and hard to visualise. The offset at any depth below the working surface is the tangent of the lean times that depth. Since the tangents involved are small, the offset is simply the ratio times the depth, and the table below is that multiplication at the depths that ordinary foundations actually use.

Pile depth	At 0.5 percent	At 1.0 percent	At 2.0 percent	At 4.0 percent
10 m	50 mm	100 mm	200 mm	400 mm
20 m	100 mm	200 mm	400 mm	800 mm
30 m	150 mm	300 mm	600 mm	1,200 mm
40 m	200 mm	400 mm	800 mm	1,600 mm
60 m	300 mm	600 mm	1,200 mm	2,400 mm

The middle column is the one to read, because one percent is the figure most bored piles in China are accepted against and the figure appears unchanged in the Australian standard and in the ICE specification for walls installed with a cable grab. At thirty metres it permits three hundred millimetres of toe offset. At sixty metres it permits six hundred. Neither of those is a lean; both are a pile that ends up somewhere other than where the drawing says, and both are compliant.

The right hand column is where the arithmetic stops being abstract. Four percent on a sixty metre pile is two point four metres. Nothing in a building tolerates that. A two point four metre shift at the toe puts the pile into the neighbouring property, out of its own pile group, past the edge of the raft and into whatever the adjacent excavation was protecting. And a pile in that condition passed the verticality clause of a European standard, because the clause asked for four percent and got it.

There is a second consequence that matters more in practice and is less often said. On one foundation, two piles can both be inside tolerance while one carries three times the toe offset of the other, simply because it is three times as long. Compliance is therefore not a state that can be compared between piles. It is a state relative to each pile’s own length, which means a schedule of readings that says compliant, compliant, compliant carries almost no information until the lengths are put beside it. That is what a taper means, and it is why a table of pass and fail is the wrong output for this measurement.

3. The Standard Already Publishes the Integral

The strongest evidence that 1 in 100 is a taper and not an angle is that one standard publishes the taper integrated, in the same table, as a separate acceptance criterion. GB 50202-2018, the Chinese acceptance standard for building foundations, gives the verticality allowance and the pile position allowance side by side, and the position allowance is not a constant. For a mud slurry bored pile with a diameter under one metre it is 70 mm plus 0.01 times H, where H is the depth from the working surface to the pile toe. That 0.01 is the same one percent.

The reason is mechanical. A pile installed at the full permitted taper drifts by 0.01 times H laterally by the time it reaches the toe, because that is what a one percent taper does over a depth H. The second term of the position allowance is exactly that drift. What is left, the 70 mm, is the surveyor’s error in setting the pile out, which is a constant and has nothing to do with how long the pile is.

The check is the residue, and it holds on two independent rows of the same table. The hand dug pile row carries a verticality of 1 in 200 and a position allowance of 50 mm plus 0.005 times H. Subtract the integral from the allowance at every depth and look at what survives.

Depth	Bored pile at 1 in 100: position allowed, and the integral inside it	Hand dug pile at 1 in 200: position allowed, and the integral inside it	What is left after the integral
10 m	170 mm, of which 100 mm	100 mm, of which 50 mm	70 mm and 50 mm
20 m	270 mm, of which 200 mm	150 mm, of which 100 mm	70 mm and 50 mm
30 m	370 mm, of which 300 mm	200 mm, of which 150 mm	70 mm and 50 mm
40 m	470 mm, of which 400 mm	250 mm, of which 200 mm	70 mm and 50 mm

The residue is 70 mm on the first row at every depth and 50 mm on the second row at every depth. Two rows, two different tapers, two different constants, and the same structure. The constant is flat, the taper term grows, and the flat part is the setting out allowance. Extend the first row to sixty metres and the allowance becomes 670 mm, of which 600 is the taper and the same 70 is the surveyor.

So the two acceptance criteria are one quantity in two forms. The verticality clause is the derivative of the lean with respect to depth, and the position clause is its integral plus a boundary condition. That is worth stating plainly, because it means the two cannot be applied as independent checks. A pile that satisfies the position allowance has, by construction, spent almost all of that allowance on the taper, and the surveyor’s 70 mm is all that is left for the setting out. A pile that satisfies both has satisfied one constraint twice.

This is a verification of the kind that is worth doing before the argument is built on it, and it is available in the same table for anyone who wants to check it. It also settles the definition. If the ratio were an angular tolerance, the position allowance would be a constant plus a term proportional to depth for some other reason. It is not. The two numbers are the same number.

4. Where the Number Is, and Where the Instrument Goes

A tilt channel measures an angle. On a pile that is exactly the right quantity, and it is in exactly the wrong place.

Follow the offset down from the platform. At the cut off, which sits at or a little below the working surface once the pile head has been broken down, a one percent lean produces ten or twenty millimetres of lateral shift. That is less than the setting out allowance and less than the argument about cover. At the toe of a thirty metre pile the same lean is three hundred millimetres. The instrument is bolted where the pile emerges. The consequence is thirty metres below it, in a place that will be covered by a cap before anyone can look.

This is a different problem from the one a manhole cover presents, and the difference is worth being precise about. On a cover the channel was the wrong kind of quantity for the motion, because a uniform lift rotates nothing. Here the channel is the right kind, the reading is correct, and the reading simply refers to a station where the quantity is at its smallest. No improvement in the sensor changes that. The fix is a second instrument class in a different place: a borehole profile tool run through the finished hole, or the sonic logging tubes cast into the shaft, both of which measure the shape along the length rather than the attitude at one end.

There is a further consequence for what a single reading means, and it is the one that most often gets lost. The head angle is a boundary condition on the profile, not the profile. A pile whose head reads zero and whose shaft curves can have a toe offset as large as one whose head carries the whole lean. The two are indistinguishable from above ground, and the second is the more dangerous of the pair, because it is the one nobody flags. A straight pile leaning by one percent is a pile with a known geometry. A pile with a straight head and a curved shaft is a pile whose geometry is unknown and whose reading is reassuring.

The published guidance for the hole side has the same shape. The verticality of a borehole is checked before the cage goes in, using an ultrasonic caliper or a borehole inclinometer logged point by point, and the acceptance question is asked of the hole rather than of the pile. That is the right station, because the hole is what the concrete will fill, and because the instrument can travel its whole length. The instrument that cannot travel is the one on the head.

5. One Hundredth of a g

Here is an identity that is exact rather than approximate, and it is the single most useful number in the subject. A tilt sensor works by resolving the gravity vector in its own frame. Any lateral acceleration adds to that vector, and the angle the instrument reports becomes the arctangent of the ratio of the lateral acceleration to g. Put that ratio at one hundredth and the arctangent is 0.5729 deg. One percent is also 0.5729 deg, because a one percent slope is a tangent of 0.01 and the angle is the same arctangent.

The two numbers are equal because they are the same expression. The tolerance for a bored pile and the apparent tilt produced by one hundredth of a g are both the arctangent of 0.01, so they coincide at every precision and always will.

The practical content is blunt. A channel on a piling rig cannot be read while the machine is working. One hundredth of a g is not a large acceleration. It is what the leader feels when the hammer is running, when a crane slews at the far end of the mat, when a crawler tracks over a timber platform, when a concrete truck passes behind the rig. Every one of those is larger than the tolerance, and several of them are an order of magnitude larger. The reading has to be taken in a still window, against a reference that is not on the machine, or both.

The two companion degrees are worth having in the same place, because they explain why the stillness is the whole problem and the sensitivity is not. The accuracy figure of 0.005 deg is eighty seven microg, and the resolution of 0.001 deg is 17.5 microg. No site produces eighty seven microg of lateral acceleration under a stationary rig. The channel is not short of sensitivity. It is short of quiet.

Two boundaries belong here. The first is that the mechanism is the one already worked through in the vibration article, which owns the averaging residual and the distinction between a direct current and an alternating one, so nothing here re-derives it. The second is the trafficked cover, which owns the same arctangent arithmetic in its noisiest form. What is new here is only the coincidence: that on a pile the tolerance itself happens to equal one hundredth of a g, so the noise threshold and the acceptance threshold are the same number rather than separated by a comfortable margin.

6. The Hardware Is Not the Limit, for the First Time

Across most of this series the datasheet has been the place where the comfortable answer comes apart. On a crane runway the zero deviation alone was one and three quarter times the whole tolerance and the maximum zero temperature drift was four and a half times it, which forced the instrument into the role of a comparator in time rather than a reference. On a pile the same instrument has room to spare.

The table below puts every published term of the ZCT2xxM-LBS-Ax-H5-460x against the two tolerances that matter, the Chinese bored pile figure of one percent and the European figure of two percent.

Term on the datasheet	Published value	Share of a 1 percent tolerance, 0.573 deg	Share of a 2 percent tolerance, 1.146 deg
Resolution	0.001 deg	0.17 percent	0.09 percent
Accuracy, tight end of the band	0.005 deg	0.87 percent	0.44 percent
Accuracy, loose end of the band	0.01 deg	1.75 percent	0.87 percent
Zero deviation	plus or minus 0.05 deg	8.73 percent	4.36 percent
Zero temperature drift, typical, over a 40 kelvin swing	0.08 deg	13.96 percent	6.98 percent
Zero temperature drift, published maximum	0.13 deg	22.69 percent	11.35 percent

The largest term on the sheet is the maximum zero temperature drift, 0.13 deg across the full minus 40 to plus 85 degrees Celsius range, and it is 22.7 percent of a one percent tolerance. The typical drift over a realistic forty kelvin swing is 14 percent, the zero deviation is 8.7 percent, the accuracy is 0.87 percent at the tight end of its band, and the resolution is 0.17 percent. Every term sits inside, and the worst one leaves three quarters of the budget untouched.

Why the reversal happens is arithmetic, and it is the same arithmetic that ran the other way in the previous article in this series. The crane runway criterion was ten millimetres over twenty metres, which is 0.0287 deg. The pile criterion is 0.5729 deg. The ratio between them is exactly twenty, and the headroom on every datasheet line improves by that same factor of twenty, because the instrument did not change at all.

So the finding is not that this sensor is better suited to a pile. It is that a specification term is worth whatever the criterion it will be read against makes it worth, and the same line that decided a runway is a rounding item on a pile. A reader who carries the runway conclusion across to a foundation will over specify by a factor of twenty and will spend the money on the wrong line of the sheet.

7. What the Specification Should Be Written On

If the datasheet is not the constraint, then the constraint is somewhere else, and on a pile it is in three places that are all about the reference rather than the sensor. Each of them produces an error that is larger than the sensor’s own, and each of them is invisible in the number that gets recorded.

The face is not the axis. A bored pile head is broken down with a hydraulic breaker, and the face that is left is a rough plane that is not square to the pile axis. A sensor seated on that face reads the face. Half a degree of out of square consumes 87.3 percent of a one percent tolerance before the pile has leaned at all, and one degree consumes 174.5 percent of it, which means a perfectly vertical pile can be rejected by its own mounting. Against the two percent European figure the same half degree is 43.6 percent and one degree is 87.3 percent.

This is the error that catches experienced crews, because the failure is invisible in the reading. A sensor that is perfectly installed on a face that is half a degree out of square reports a steady half degree, which looks exactly like a pile leaning at half of its allowance. The crew records it, the pile passes, and the real geometry is unknown. What the installation procedure has to produce is the pile axis, established by a sleeve that is set before the cut, by a jig that references the shaft below the cut, or by a survey of the shaft in two directions, and the record has to carry a statement of which method was used. An angle without that statement is an angle about an unknown line.

One axis has a blind direction. A lean at an angle to the sensor axis reads the lean times the cosine of that angle, so the reading is always either correct or too small and never too large. The table is that cosine at the angles a lean can actually sit at.

Angle between the lean and the sensor axis	What the channel reads	A lean that reports as 1.00 percent
0 deg	100 percent of the lean	1.00 percent
30 deg	86.6 percent	1.15 percent
45 deg	70.7 percent	1.41 percent
60 deg	50.0 percent	2.00 percent
75 deg	25.9 percent	3.86 percent
90 deg	zero	no value of the lean reaches it

The third column is the one worth keeping, because it inverts the table into the question a reader actually has: given a reading of exactly the acceptance value, how bad could the pile be. The answer is worst at forty five degrees, where a lean of 1.41 percent reads as 1.00 percent, which is a pile forty one percent over the tolerance passing as though it were the textbook example. The intuition points the other way here. Most people expect the dangerous region to be near ninety degrees, where the reading collapses towards zero and somebody notices that the instrument has stopped moving. It is not. At ninety degrees the reading is so wrong that it is self announcing. At forty five it sits at the acceptance value and looks entirely plausible, and the channel therefore fails quietly at half past the diagonal rather than loudly at the perpendicular.

The axis has to be defined before it can be measured. Verticality is a statement about gravity, and gravity is what the channel measures, so the instrument suits the definition. But the pile is a rough cylinder and its axis is a convention: the axis of the cage, of the casing, of the drilled hole, or of the finished shaft. A tolerance of one percent on the hole is not a tolerance of one percent on the cage, because the cage is lowered into a hole that is already leaning.

This third one is a convention rather than an error, and it is the one that decides whether two parties are arguing about the same thing. Two documents can both say one percent and mean two different lines, and the difference between the hole axis and the cage axis at a depth of thirty metres is not a rounding term. It is the difference between an acceptance that was met and an acceptance that was met against a different object, and no amount of instrument stability resolves it. The only defence is to name the line in the record.

8. The Consequence Is Not Bending

It is tempting to treat a taper as a bending problem, and the arithmetic says that reading is wrong. Suppose the head were rotationally fixed by the cap and the toe were fixed in the soil. A lean of theta between two fixed ends is a chord rotation, and the elastic moment it induces is six EI theta over L, or three EI theta over L with one end pinned. Both are easy to compute and both give an answer that no foundation engineer recognises.

Pile diameter	Cracking moment at 3 MPa	Moment at 30 m, both ends fixed, 1 percent lean	Times the cracking moment
0.6 m	64 kN m	382 kN m	6.0
0.8 m	151 kN m	1,206 kN m	8.0
1.0 m	295 kN m	2,945 kN m	10.0
1.2 m	509 kN m	6,107 kN m	12.0
1.5 m	994 kN m	14,910 kN m	15.0

A one metre pile at thirty metres, fixed at both ends, would carry about 2,945 kN m. Its cracking moment, at a concrete tensile strength of 3 MPa, is about 295 kN m. The elastic demand is ten times the cracking moment; with one end pinned it is five times. Either way, a pile installed at the full permitted taper and rotationally restrained at both ends would be cracked along its length, and nobody believes that piles at the edge of the verticality tolerance are routinely cracked.

The resolution is that neither restraint exists in the form the formula assumes. A cap is not a moment connection to a rough pile head, and the soil at depth yields laterally long before it delivers a fixed toe. The moment that actually develops is the one the soil is willing to pay for, which is a fraction of the elastic value, and it belongs to the lateral load design rather than to the verticality acceptance. The elastic number is useful only as a demonstration that the tolerance cannot be a structural limit, because if it were, the tolerance and the strength check would contradict each other by an order of magnitude.

The transverse component is the other candidate and it is also small. An axial load on a pile leaning at theta has a component P sine theta, which at a one percent lean is almost exactly one percent of the axial load: 50 kN on a 5,000 kN pile, 100 kN on a 10,000 kN one. The soil takes most of it, the group takes some, and the cap takes the rest. It is not nothing, and it is not why the tolerance exists.

What the tolerance exists for is fit, and the fit argument is entirely geometric. A leaning pile has to pass through a cap whose plan position is fixed by the column above, so the head has to land inside the cap with cover, which is the 70 mm setting out allowance doing its work. It has to keep clear of its neighbours, and the clearance that matters is at depth, where the taper has accumulated, not at the surface where the spacing was drawn. It has to stay inside the property line and outside the zone that a future excavation will disturb. And where the pile is end bearing, its toe has to land on the stratum it was designed for, which a three hundred millimetre offset can miss on a layered site.

None of those checks appear in the verticality clause, and all of them are the reason the clause exists. The clause is a proxy. The quantity it proxies is a distance, and the distance is thirty metres from where the clause is checked.

9. What to Specify, and What Not to Claim

There are three legitimate uses of an angle channel in a pile foundation, and they need three different specifications. The table sets them out.

Use	What it is	What it answers
Control during installation	A channel on the leader, the casing or the mandrel, read in a still window	Whether to keep driving or to stop and reposition, in the first few metres
One reading at cut off	A channel on the pile head, with a stated reference for the axis	The as built inclination that the cap has to accept
After the cap is cast	Nothing on the pile	The attitude of the cap and the structure, which is a different subject

The first is the one with the most value and the least resemblance to monitoring. Before the pile is locked into the ground, a channel on the leader or the casing is a control input: it says whether to keep driving or to stop and reposition, and its worth is concentrated in the first few metres because that is the only window in which correction is possible at all. This is the first subject in this series where the measurement closes a loop on a machine rather than describing a structure, and the specification follows from that. Sample fast, alarm locally, no cloud link, no trend, and a mounting that survives the hammer.

The second is a single reading taken once. It does not need a network, a heartbeat, a battery life in years or a platform. It needs a reference, a date, and a photograph of how the sensor was seated, because the reference is where the error lives and the photograph is the only evidence of it afterwards. A reading without the photograph is a number about an unknown line.

The third is none, and saying so is more useful than proposing a scheme. Once the cap is cast, a tilt channel on a pile head is not measuring the pile. It is measuring the cap, and above the cap it is measuring the structure, and both of those belong to other articles in this series. The pile’s geometry was fixed on the day the concrete set. This is the first subject in the series where the quantity of interest has no trend at all, so a monitoring scheme built on a trend is a scheme built on nothing.

Four limits follow and each of them is a statement about what the channel is not. It cannot produce a profile, because a profile is an integral and the channel reports a boundary condition. It cannot see a lean perpendicular to its axis, and at forty five degrees it under reports by twenty nine percent of the lean, which is enough for a pile forty one percent over the tolerance to report exactly at it. It cannot separate the pile from its reference face, so a mounting error enters the reading one for one. And it cannot measure anything while the hammer is running, because one hundredth of a g is both the noise threshold and the tolerance.

On the company side, no pile installation deployment is claimed here, and the reader should not infer one from the presence of a datasheet in this article. The nearest published work in this family is the borehole side, where in place inclinometers are used to log lateral displacement along a vertical line, which is the same instrument class that would produce a pile profile and a different purpose from the one described above. The arithmetic in this article is reconstructed from public acceptance standards and from the published datasheet, and it is a design argument rather than a field record.

Frequently Asked Questions

1. Is 1 in 100 a verticality tolerance or a position tolerance? It is the same quantity written twice. The verticality clause states the taper, and the position clause states the taper integrated over the depth plus a constant for setting out. In GB 50202-2018 the bored pile row is 1 in 100 for verticality and 70 mm plus 0.01 times the depth for position, and subtracting the integral leaves exactly 70 mm at every depth. A pile cannot satisfy one of those clauses without spending the other, so they are not independent checks.
2. Can a tilt sensor measure pile verticality? It measures the inclination of whatever it is bolted to, which is useful in two places and misleading in a third. On the leader or casing during installation it is a control input. On the pile head at cut off it records the as built inclination, provided the reference axis is established and documented. After the cap is cast it measures the cap and the structure, not the pile, because the pile’s geometry stopped changing when the concrete set.
3. Why is the toe offset so much larger than the head offset? Because the offset grows linearly with depth below the datum while the angle stays constant. At one percent a pile shifts ten millimetres for every metre of depth, so a head a metre below the platform moves ten millimetres and a toe at thirty metres moves three hundred. The instrument sits where the number is smallest and the consequence sits where nobody can reach it, which is why the profile needs a borehole instrument rather than a better sensor on the head.
4. Which tolerance should a specification use? Whichever one the foundation’s geometry can absorb, stated as a taper and checked against the toe offset it permits. The published range is a factor of ten, from 0.4 percent to 4 percent, and four percent on a sixty metre pile permits 2.4 metres of toe offset, which no building can take. Converting the chosen ratio into millimetres at the actual pile length, and comparing that with the cap size, the group spacing at depth and the property line, is the check that decides the number rather than the table.

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